Towards a Solution of Higher Airy Structures in Terms of Tau Functions and Matrix Models Article Swipe

The first part of this thesis presents an exposition of some of the links between integrability, matrix models and Airy structures. We first introduce the notion of an integrable hierarchy and a tau function. We then probe tau functions further by considering matrix integrals. These are typically a good source of W-algebra representations that annihilate tau functions. We subsequently investigate such W-constraints from the point of view of higher Airy structures introduced in [1]. These constraints depend on two parameters, r and s. In the second part of this thesis, we investigate further the r-KW and r-BGW tau functions using an external field matrix model. Through the works of various authors, the r-KW tau function is well understood from these different perspectives. The r-BGW tau function is far less well understood, however. We speculate on the form of the spectral curve for the r-BGW tau function, implying the corresponding Airy structure is given by s=r-1. We subsequently present an explicit calculation of the Virasoro constraints for the 2-BGW tau function using Ward identities for an external field matrix model. We then match this with the (2,1) Airy structure. While this result is already known, the exact details have never been shown. We then generalise this calculation to the 3-BGW tau function by constructing the lowest order cubic mode of the W-constraints for an external field model. While it is still unclear if this exactly matches with the lowest cubic mode from the (3,2) Airy structure, we believe that these two modes do indeed match, after some minor modifications.