Boundaries, States and Cohomology in Three-Dimensional N = 4 Theories

This thesis studies geometric and algebraic aspects of 3d N = 4 theories. We first focus on 3d N = 4 gauge theories compactified on an elliptic curve, and provide physical realisations of the equivariant elliptic cohomology of symplectic resolutions, and recent constructions therein. The Berry connection for supersymmetric ground states in the presence of mass parameters and flat connections for flavour symmetries is analysed, resulting in a natural construction of the equivariant elliptic cohomology variety of the Higgs branch. Supersymmetric boundary conditions are investigated in this set-up. From an analysis of boundary 't Hooft anomalies, their boundary amplitudes are demonstrated to represent equivariant elliptic cohomology classes. We then investigate two distinguished classes of N = (2,2) boundary conditions, each in 1-1 correspondence with the set of isolated massive vacua, known as exceptional Dirichlet and enriched Neumann. The former mimic isolated vacua at infinity in the presence of real mass and FI parameters. The two classes are further shown to be exchanged under mirror symmetry, via collision with a mirror symmetry interface. By computing boundary amplitudes, the enriched Neumann boundary conditions reproduce the elliptic stable envelopes of Aganagic-Okounkov, and the mirror symmetry interface the mother function in equivariant elliptic cohomology. Finally, correlation functions of Janus interfaces for varying mass parameters are considered, recovering the chamber R-matrices of elliptic integrable systems. We then study the factorisation of partition functions of N = 4 theories on closed 3-manifolds (such as the superconformal index, twisted index and S³ partition function) in terms of S¹ x HS² partition functions. We demonstrate the latter, equipped with exceptional Dirichlet boundary conditions, realise this factorisation exactly, and can be unambiguously defined and computed using supersymmetric localisation. We show certain limits of these hemisphere partition functions yield characters of lowest weight Verma modules over the quantised Higgs and Coulomb branch chiral rings. This leads to expressions for the closed 3-manifold partition functions 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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2022

Author dc:creator

• Zhang, Daniel

Advisor dc:contributor.advisor

• Dorey, Nicholas

• algebraic geometry
• elliptic cohomology
• gauge theory
• mathematical physics
• quantum field theory
• representation theory
• supersymmetry

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/b76e8944-a97f-46c5-b81d-2121f8c3b349/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

Author Identifier

0000-0001-9180-2436

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/349594