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

Abstract

We consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure $\mathcal{N}=2$ SYM with gauge group $SU(3)$ in Nekrasov-Shatashvili limit of $\Omega$-background. We show that this is the same differential equation that emerges in the context of Ordinary Differential Equation/Integrable Models (ODI/IM) correspondence for $2d$ $A_2$ Toda CFT with central charge $c=98$. We derive the corresponding $QQ$ and related $TQ$ functional relations and establish the asymptotic behaviour of $Q$ and $T$ functions at small instanton parameter $q\to 0$. 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 parametrised by the vector multiplet scalar VEVs $⟨\text{tr}{\varphi }^2⟩$ and $⟨\text{tr}{\varphi }^3⟩$ 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.

arXiv comment20 pages, minor corrections, references added

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