# YTF 20

Dec 15 – 16, 2020
Centre for Particle Theory, Durham
Europe/London timezone

## Boundaries, Vermas, and Factorisation

Dec 16, 2020, 2:30 PM
30m
Teams (Centre for Particle Theory, Durham)

### Teams

#### Centre for Particle Theory, Durham

IPPP & Dept. Mathematical Sciences Durham University South Road Durham DH1 3LE

### Speaker

Daniel Zhang (University of Cambridge)

### Description

I will discuss a recent work on the factorisation of closed 3-manifold partition functions and indices of 3d $\mathcal{N}=4$ gauge theories. The building blocks are hemisphere partition functions equipped with a class of UV $\mathcal{N}=(2,2)$ boundary conditions that mimic the presence of isolated vacua at infinity. Via the state-operator correspondence, these count local operators supported on a $(2,2)$ boundary condition on a plane. A subset of these operators are boundary Higgs and Coulomb branch operators, which form lowest weight Verma modules over the quantised bulk Higgs and Coulomb branch chiral rings. We show that certain limits of the hemisphere partition functions compute their characters. We find that the equivariant supersymmetric Casimir energy encodes the boundary ’t Hooft anomaly, and also plays the role of highest weights. Applying these results to factorisation then leads to various “IR formulae” for partition functions on closed 3-manifolds in terms of these Verma characters.

Length of talk 15-25 minutes Yes No Yes

### Primary author

Daniel Zhang (University of Cambridge)

### Presentation materials

 Daniel Zhang YTF Talk.pdf Dan-Zhang.mp4