The talk will consist of two parts. In the first, longer, part I will review the basics of quantum toroidal algebras and their representation theory focusing on the simplest example of type gl(1). I will also discuss the identification between the representations of the algebra and branes of Type IIB string theory, how brane interactions manifest themselves in this setup and lead to integrability.

In the second part, if time permits, I will introduce new brane configurations inspired by algebraic considerations which I call spiralling branes. I will compute the corresponding partition functions and show that they coincide with the K-theoretic vertex function of Nekrasov and Okounkov and the magnificent four partition function recently introduced by Nekrasov, elucidating the algebraic meaning of both of them.