On Cohomological Algebras in Supersymmetric Quantum Field Theories
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Date
2019-08-16
Authors
Ishtiaque, Nafiz
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Publisher
University of Waterloo
Abstract
In this thesis we compute certain supersymmetric subsectors of the algebra of observables in some QFTs and demonstrate an application of such computation in checking an instance of Holographic duality. Computing the algebra of observables beyond perturbative approximation in weakly coupled field theories is far from a tractable problem. In some special, yet interesting large classes of supersymmetric theories, supersymmetry can be used to extract exact nonperturbative information about certain subsets of observables. This is an old idea which we advance in this thesis by introducing new techniques of computations, computing certain observalbes for the first time, and reproducing earlier results about some other observables. We also propose a new toy model of holographic duality involving topological/holomorphic theories, demonstrating the power of exact computations in supersymmetric subsectors. To be more specific, the subject of this thesis includes the following:
1. Computing the algebra of chiral and twisted chiral operators in 2d N=(2,2) theories -- while these algebras were previously known, we demonstrate how they can be computed using relatively modern techniques of supersymmetric localization.
2. Computing the chiral rings of 4d N=2 SCFTs -- we compute this algebra for the first time. We use the same method of supersymmetric localization that we use in the 2d case.
3. Computing the algebra of operators on a defect in the topological 2d BF theory, along with its holographic dual. This is a new toy model of holographic duality set in the world of 6d topological string theory. We also argue that this setup is in fact a certain supersymmetric subsector of the holographic duality involving 4d N=4 SYM theory and its 10d supergravity dual -- both involving some defects.
In order to be able to discuss these different theories in different dimensions with different symmetries without sounding disparate and ad hoc, we employ the language of cohomological algebra. Since this is perhaps not a language most commonly used in the standard physics literature, we would like to emphasize that this is not a novel idea, it is merely a convenient thematic and linguistic umbrella that covers all the topics of this thesis. In the BV formulation of a QFT, the algebra of observables is presented as the cohomology algebra of a certain complex consisting of fields and anti-fields. In this language restriction to supersymmetric subsectors correspond to modifying the BV differential by the addition of the relevant supersymmetry generator. We simply refer to this modification as reduction to cohomology (with respect to the choice of supersymmetry).
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Keywords
quantum field theory, supersymmetry, bps algebra, localization, holography