Gauge theories and Painlevé equations: a new perspective

In this thesis we study the linear Lax problems in x and t associated to the Painlevé V I and V equations when the Painlevé solution y(t) approaches a zero. In this limit the problem in x for the Painlevé V I reduces to the Heun Equation, while that for the Painlevé V to the Confluent Heun Equation. This equations appear as Nekrasov-Shatashvili quantisation/deformations (Ω-ackground with ε1 = ℏ, ε2 = 0) of Seiberg-Witten differentials for SU(2) N = 2 super Yang-Mills gauge theory with number of flavours Nf = 4 and Nf = 3, respectively. These new results reinforce the novel idea that Painlevé equations are each related to a different matter theory, which is actually the same as the well-estabilished Painlevé gauge correspondence with another deformation, the Self-Dual limit of the Ω-background (ε1 = −ε2 = ℏ). Furthermore, in order to complete this work on these correspondences, we compute recursion relations for the short-distance expansion of the τ -function associated to the Painlevé III3, III2 and III1 equations without introducing conformal blocks through the AGT correspondence, obtaining a recursion relation for the desired instanton contribution to the 4-dimensional Nekrasov partition function in the Self-Dual Ω-background for Nf = 0, 1, 2. Then we show how to go from the expansion for the Painlevé III1 τ - function to the one for the III2 and III3 with a double scaling limit. For a future perspective we aim to strengthen the relation between this two different limits of the Ω-background through the Nakajima–Yoshioka blow up equations.