In recent years there has been an increased interest in exploring the connections between
various disciplines of mathematics and theoretical physics such as representation theory
algebraic geometry quantum field theory and string theory. One of the challenges of modern
mathematical physics is to understand rigorously the idea of quantization. The program of
quantization by branes which comes from string theory is explored in the book. This open
access book provides a detailed description of the geometric approach to the representation
theory of the double affine Hecke algebra (DAHA) of rank one. Spherical DAHA is known to arise
from the deformation quantization of the moduli space of SL(2 C) flat connections on the
punctured torus. The authors demonstrate the study of the topological A-model on this moduli
space and establish a correspondence between Lagrangian branes of the A-model and DAHA modules.
The finite-dimensional DAHA representations are shown to be in one-to-one correspondence with
the compact Lagrangian branes. Along the way the authors discover new finite-dimensional
indecomposable representations. They proceed to embed the A-model story in an M-theory brane
construction closely related to the one used in the 3d 3d correspondence as a result modular
tensor categories behind particular finite-dimensional representations with PSL(2 Z) action are
identified. The relationship of Coulomb branch geometry and algebras of line operators in 4d N
= 2* theories to the double affine Hecke algebra is studied further by using a further
connection to the fivebrane system for the class S construction. The book is targeted at
experts in mathematical physics representation theory algebraic geometry and string theory.
This is an open access book.
