BPS states in the Minahan-Nemeschansky $ E_7 $ theory

Abstract:In this talk, I will talk about the joint work on BPS index of Minahan-Nemeschansky $E_7$ theory with Lotte Hollands and Andrew Neitzke. BPS index is the numerical DT invariant in Kontsevich-Soibelman’s paper. We use the technique of spectral network to study the BPS index in $4d$ $N=2$ class S theory. For MN $E_7$ theory we are able to calculate the BPS index for $n(\gamma_1)$ and $n’(\gamma_1+\gamma_2)$ to some order. Our result provides a proof of enhancement of flavor symmetry from the manifest $SU(4)\times SU(4)\times SU(2)$ from class S realization to $E_7$ by showing the BPS index with flavor symmetry information included are representations of $E_7$.