T, Q and periods in SU(3) N $$ \mathcal{N} $$ = 2 SYM

Abstract We consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure N $$ \mathcal{N} $$ = 2 SYM with gauge group SU(3) in Nekrasov- Shatashvili limit of Ω-background. We show that this is the same differential equation that emerges in the context of Ordinary Differential Equation/Integrable Models (ODE/IM) correspondence for 2d A 2 Toda CFT with central charge c = 98. We derive the corresponding QQ and related T Q functional relations and establish the asymptotic behaviour of Q and T functions at small instanton parameter q → 0. Moreover, numerical integration of the Floquet monodromy matrix of the differential equation leads to evaluation of the A-cycles a 1,2,3 at any point of the moduli space of vacua parametrized by the vector multiplet scalar VEVs ( tr 𝜙2 ) and ( tr 𝜙3 ) even for large values of q which are well beyond the reach of instanton calculus. The numerical results at small q are in excellent agreement with instanton calculation. We conjecture a very simple relation between Baxter’s T -function and A-cycle periods a 1,2,3, which is an extension of Alexei Zamolodchikov’s conjecture about Mathieu equation.