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In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter $q$ for ${\cal N}=2$ SYM with up to four antifundamental hypermultiplets in NS limit of $\Omega$ background. We propose a new method for calculating the period and demonstrate its efficiency by explicit calculations. The new way of doing instanton counting is more advantageous compared to known standard techniques and allows to reach substantially higher order terms with less effort. This approach is applied for the pure case as well as for the case with several hypermultiplets. We also investigate a numerical method for deriving the $A$-cycle period valid for arbitrary values of $q$. Analyzing large $q$ asymptotic we get convincing agreement with an analytic expression deduced from a conjecture by Alexei Zamolodchikov in a different context., Comment: 28 pages, 6 figures, some clarifications and citations added, published version

We consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure $$ \mathcal{N} $$ 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 A2 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 a1,2,3 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 a1,2,3, which is an extension of Alexei Zamolodchikov’s conjecture about Mathieu equation.

Analytic expressions for the two dimensional N = 1 $$ \mathcal{N}=1 $$ SLFT blocks in the light semi-classical limit are found for both Neveu-Schwarz and Ramond sectors. The calculations are done by using the duality between SU(2) N = 2 $$ \mathcal{N}=2 $$ super-symmetric gauge theories living on R 4 /Z 2 space and two dimensional N = 1 $$ \mathcal{N}=1 $$ super Liouville field theory. It is shown that in the light asymptotic limit only a restricted set of Young diagrams contributes to the partition function. This enables us to sum up the instanton series explicitly and find closed expressions for the corresponding N = 1 $$ \mathcal{N}=1 $$ SLFT four point blocks in the light asymptotic limit.