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

TL;DR

This paper connects a third order differential equation from deformed Seiberg-Witten theory in $SU(3)$ ${ m N}=2$ SYM to integrable models, deriving functional relations, analyzing asymptotics, and numerically evaluating moduli space periods beyond instanton calculus.

Contribution

It establishes a novel link between the differential equations in ${ m N}=2$ SYM and integrable models, deriving functional relations and numerically evaluating periods across the moduli space.

Findings

Derived $QQ$ and $TQ$ relations for the differential equation.

Numerically evaluated $A$-cycle periods at large $q$ beyond instanton methods.

Found agreement between numerical results and instanton calculations at small $q$.

Abstract

We consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure ${\cal N}=2$ 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 $q \rightarrow 0$. 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 $\langle \textbf{tr}\,\phi^2\rangle$ and $\langle \textbf{tr}\,\phi^3\rangle$ 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.