Francisco Correa

Geometric pathways: From integrable models to black hole physics

A short course on integrable systems, Chern-Simons theory, and their role in black hole physics

In this lecture series, I will explore how integrable systems and their defining properties can be given a natural geometric interpretation within the framework of black hole physics. The first lecture introduces foundational concepts in integrable models, including the Lax pair formalism, the zero curvature condition, and the appearance of hyperelliptic curves in certain nonlinear integrable hierarchies. The second lecture focuses on Chern-Simons theory, with a detailed discussion of how (2+1)-dimensional gravity with a negative cosmological constant can be reformulated in terms of two

s l (2, R)

Chern-Simons connections. In the final lecture, we synthesize the previous ideas to show how the zero curvature condition arising in Chern-Simons gravity provides a geometric language to reinterpret various integrable systems. We will demonstrate how several features of integrable models discussed earlier emerge naturally in the context of integrable black hole solutions and their associated asymptotic symmetries.