Geometric aspects of three dimensional N = 4 gauge theories
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Authors
Crew, Samuel
Advisors
Dorey, Nick
Date
2021-01-01Awarding Institution
University of Cambridge
Qualification
Doctor of Philosophy (PhD)
Type
Thesis
Metadata
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Crew, S. (2021). Geometric aspects of three dimensional N = 4 gauge theories (Doctoral thesis). https://doi.org/10.17863/CAM.80877
Abstract
We study geometric aspects of three dimensional N = 4 gauge theories. We focus mainly
on the factorisation property of supersymmetric partition functions of these theories and
introduce hemisphere blocks that precisely realise the factorisation. We define these blocks
as UV partition functions on a hemisphere S1 ×H2 with an exceptional Dirichlet boundary
condition and demonstrate that the resulting object is determined by the enumerative
geometry of the Higgs branch.
The partition function on the hemisphere is closely related to a half-index that counts
local operators of the theory on a flat spacetime with boundary. In this context, we show
that the hemisphere blocks realise characters of lowest weight Verma modules of the Higgs
and Coulomb branch chiral rings acting on boundary local operators.
We study the geometric interpretation of the twisted index factorisation in particular
and demonstrate a relationship between the twisted index and the Hilbert series of a 3d
N = 4 theory. We then use factorisation to provide a novel geometric expression for the
Coulomb branch Hilbert series in terms of invariants of moduli spaces of quasimaps to the
Higgs branch.
Finally, we apply these ideas to a particularly rich example of a non-abelian gauge theory
with adjoint matter whose Higgs branch coincides with a moduli space of instantons. We
compute hemisphere blocks for the theory and explicitly recover Verma module characters of
the Coulomb branch chiral ring. In this example, the blocks have interesting combinatorial
content and can be related to generating functions of reverse plane partitions—we discuss
the interpretation of 3d mirror symmetry in this context. We also study line operators
in this theory and show that half indices in the presence of a line operator exhibit an
integrable structure. Along the way we find interesting connections between the twisted
index gluing of hemisphere blocks and related calculations in topological string theory.
Keywords
quantum field theory
Identifiers
This record's DOI: https://doi.org/10.17863/CAM.80877
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