Boundaries, Vermas and factorisation
Mathew Bullimore, Samuel Crew, Daniel Zhang
Abstract
A bstract We revisit the factorisation of supersymmetric partition functions of 3d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 gauge theories. The building blocks are hemisphere partition functions of a class of UV $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = (2 , 2) boundary conditions that mimic the presence of isolated vacua at infinity in the presence of real mass and FI parameters. These building blocks can be unambiguously defined and computed using supersymmetric localisation. We show that certain limits of these hemisphere partition functions coincide with characters of lowest weight Verma mod- ules over the quantised Higgs and Coulomb branch chiral rings. This leads to expressions for the superconformal index, twisted index and S 3 partition function in terms of such characters. On the way we uncover new connections between boundary ’t Hooft anomalies, hemisphere partition functions and lowest weights of Verma modules.