{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/260821"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/260821","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"The geometry and representation theory of superconformal quantum mechanics","abstract":"We study aspects of the quantum mechanics of nonlinear $\\sigma$-models with superconformal invariance. The connection between the differential geometry of the target manifold and symmetries of the quantum mechanics is explored, resulting in a classification of spaces admitting $\\mathcal{N}=(n,n)$ superconformal invariance with $n=1,2,4$. We construct the corresponding superalgberas $\\mathfrak{su}(1,1|1),~\\mathfrak{u}(1,1|2)$ and $\\mathfrak{osp}(4^*|4)$ explicitly. The low-energy dynamics of Yang-Mills instantons is an example of the latter and arises naturally in the discrete light-cone quantisation (DLCQ) of certain superconformal field theories. In particular, we study in some detail the quantum mechanics arising in the DLCQ of the six-dimensional (2,0) theory and four-dimensional $\\mathcal{N}=4$ SUSY Yang-Mills. In the (2,0) case we carry out a detailed study of the representation theory of the light-cone superalgebra $\\mathfrak{osp}(4^*|4)$. We give a complete classification of the unitary irreducible representations and their branching at the unitarity bound, and use this information to construct the superconformal index for $\\mathfrak{osp}(4^*|4)$. States contribute to the index if and only if they are in the cohomology of a particular supercharge, which we identify as the $L^2$ Dolbeault cohomology of instanton moduli space with values in a real line bundle. In the SUSY Yang-Mills case the target space is the Coulomb branch of an elliptic quiver gauge theory, and as such is a scale-invariant special Kähler manifold. We describe a new type of $\\sigma$-model with $\\mathcal{N}=(4,4)$ superconformal symmetry and $U(1)\\times SO(6)$ R-symmetry which exists on any such manifold. These models exhibit $\\mathfrak{su}(1,1|4)$ invariance and we give an explicit construction of the superalgebra in terms of known functions. Consideration of the spectral problem for the dilatation operator in these models leads to a deformation which we interpret, via an extension of the moduli space approximation, as an anti-self-dual spacetime magnetic field coupling to the topological instanton current.","abstract_html":"We study aspects of the quantum mechanics of nonlinear <span class=\"etd-inline-math\">&sigma;</span>-models with superconformal invariance. The connection between the differential geometry of the target manifold and symmetries of the quantum mechanics is explored, resulting in a classification of spaces admitting $\\mathcal{N}=(n,n)$ superconformal invariance with $n=1,2,4$. We construct the corresponding superalgberas $\\mathfrak{su}(1,1|1),~\\mathfrak{u}(1,1|2)$ and <span class=\"etd-inline-math\">\\mathfrak{osp}(4<sup>*</sup>|4)</span> explicitly. The low-energy dynamics of Yang-Mills instantons is an example of the latter and arises naturally in the discrete light-cone quantisation (DLCQ) of certain superconformal field theories. In particular, we study in some detail the quantum mechanics arising in the DLCQ of the six-dimensional (2,0) theory and four-dimensional $\\mathcal{N}=4$ SUSY Yang-Mills. In the (2,0) case we carry out a detailed study of the representation theory of the light-cone superalgebra <span class=\"etd-inline-math\">\\mathfrak{osp}(4<sup>*</sup>|4)</span>. We give a complete classification of the unitary irreducible representations and their branching at the unitarity bound, and use this information to construct the superconformal index for <span class=\"etd-inline-math\">\\mathfrak{osp}(4<sup>*</sup>|4)</span>. States contribute to the index if and only if they are in the cohomology of a particular supercharge, which we identify as the <span class=\"etd-inline-math\">L<sup>2</sup></span> Dolbeault cohomology of instanton moduli space with values in a real line bundle. In the SUSY Yang-Mills case the target space is the Coulomb branch of an elliptic quiver gauge theory, and as such is a scale-invariant special Kähler manifold. We describe a new type of <span class=\"etd-inline-math\">&sigma;</span>-model with $\\mathcal{N}=(4,4)$ superconformal symmetry and $U(1)\\times SO(6)$ R-symmetry which exists on any such manifold. These models exhibit $\\mathfrak{su}(1,1|4)$ invariance and we give an explicit construction of the superalgebra in terms of known functions. Consideration of the spectral problem for the dilatation operator in these models leads to a deformation which we interpret, via an extension of the moduli space approximation, as an anti-self-dual spacetime magnetic field coupling to the topological instanton current.","abstract_has_math":true,"creators":["Singleton, Andrew John"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-06-28","date_published":"2016-06-28","updated_at":"2026-07-22T22:24:20Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9171c46a-d3c2-4d02-8dd6-f91a2f5f5874/download","https://www.rioxx.net/licenses/all-rights-reserved/","https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9866eafd-05af-4ac4-ad16-a412e39fb02c/download"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.275","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Singleton, Andrew John"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2016-06-28"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/260821"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9171c46a-d3c2-4d02-8dd6-f91a2f5f5874/download","https://www.rioxx.net/licenses/all-rights-reserved/","https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9866eafd-05af-4ac4-ad16-a412e39fb02c/download"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.275"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/d34ab446-7d05-448f-a72b-ea7eb9ac59a5/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We study aspects of the quantum mechanics of nonlinear $\\sigma$-models with superconformal invariance. The connection between the differential geometry of the target manifold and symmetries of the quantum mechanics is explored, resulting in a classification of spaces admitting $\\mathcal{N}=(n,n)$ superconformal invariance with $n=1,2,4$. We construct the corresponding superalgberas $\\mathfrak{su}(1,1|1),~\\mathfrak{u}(1,1|2)$ and $\\mathfrak{osp}(4^*|4)$ explicitly. The low-energy dynamics of Yang-Mills instantons is an example of the latter and arises naturally in the discrete light-cone quantisation (DLCQ) of certain superconformal field theories. In particular, we study in some detail the quantum mechanics arising in the DLCQ of the six-dimensional (2,0) theory and four-dimensional $\\mathcal{N}=4$ SUSY Yang-Mills. In the (2,0) case we carry out a detailed study of the representation theory of the light-cone superalgebra $\\mathfrak{osp}(4^*|4)$. We give a complete classification of the unitary irreducible representations and their branching at the unitarity bound, and use this information to construct the superconformal index for $\\mathfrak{osp}(4^*|4)$. States contribute to the index if and only if they are in the cohomology of a particular supercharge, which we identify as the $L^2$ Dolbeault cohomology of instanton moduli space with values in a real line bundle. In the SUSY Yang-Mills case the target space is the Coulomb branch of an elliptic quiver gauge theory, and as such is a scale-invariant special Kähler manifold. We describe a new type of $\\sigma$-model with $\\mathcal{N}=(4,4)$ superconformal symmetry and $U(1)\\times SO(6)$ R-symmetry which exists on any such manifold. These models exhibit $\\mathfrak{su}(1,1|4)$ invariance and we give an explicit construction of the superalgebra in terms of known functions. Consideration of the spectral problem for the dilatation operator in these models leads to a deformation which we interpret, via an extension of the moduli space approximation, as an anti-self-dual spacetime magnetic field coupling to the topological instanton current."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["7c3d2baf9acefde6b07ad115ddde4e3c","87eda9de84448d1f82354d60eee3eb5f","89d8176b6396945b84ededaa89aeced0"]},{"key":"dc:title","label":"Title","values":["The geometry and representation theory of superconformal quantum mechanics"]}]}],"canonical_facts":{"dc:creator":["Singleton, Andrew John"],"dc:date.issued":["2016-06-28"],"dc:description.abstract":["We study aspects of the quantum mechanics of nonlinear $\\sigma$-models with superconformal invariance. The connection between the differential geometry of the target manifold and symmetries of the quantum mechanics is explored, resulting in a classification of spaces admitting $\\mathcal{N}=(n,n)$ superconformal invariance with $n=1,2,4$. We construct the corresponding superalgberas $\\mathfrak{su}(1,1|1),~\\mathfrak{u}(1,1|2)$ and $\\mathfrak{osp}(4^*|4)$ explicitly. The low-energy dynamics of Yang-Mills instantons is an example of the latter and arises naturally in the discrete light-cone quantisation (DLCQ) of certain superconformal field theories. In particular, we study in some detail the quantum mechanics arising in the DLCQ of the six-dimensional (2,0) theory and four-dimensional $\\mathcal{N}=4$ SUSY Yang-Mills. In the (2,0) case we carry out a detailed study of the representation theory of the light-cone superalgebra $\\mathfrak{osp}(4^*|4)$. We give a complete classification of the unitary irreducible representations and their branching at the unitarity bound, and use this information to construct the superconformal index for $\\mathfrak{osp}(4^*|4)$. States contribute to the index if and only if they are in the cohomology of a particular supercharge, which we identify as the $L^2$ Dolbeault cohomology of instanton moduli space with values in a real line bundle. In the SUSY Yang-Mills case the target space is the Coulomb branch of an elliptic quiver gauge theory, and as such is a scale-invariant special Kähler manifold. We describe a new type of $\\sigma$-model with $\\mathcal{N}=(4,4)$ superconformal symmetry and $U(1)\\times SO(6)$ R-symmetry which exists on any such manifold. These models exhibit $\\mathfrak{su}(1,1|4)$ invariance and we give an explicit construction of the superalgebra in terms of known functions. 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