{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/125667"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/125667","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Embedding degree of freedom, integrable charges and extended phase space","abstract":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-08-01","abstract_html":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-08-01","abstract_has_math":false,"creators":["Pai, Pin-Chun"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Physics","degree_department":null,"school":null,"contributors":["Leigh, Robert","Faulkner, Thomas","Fradkin, Eduardo","Hughes, Taylor"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-06-18","date_published":"2024-06-18","updated_at":"2026-07-22T22:25:02Z","subjects":["Extended Phase Space"],"languages":["en","eng"],"rights":["Copyright 2024 Pin-Chun Pai"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/125667","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Leigh, Robert","Faulkner, Thomas","Fradkin, Eduardo","Hughes, Taylor"]},{"key":"dc:creator","label":"Author","values":["Pai, Pin-Chun"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2024-06-18","2024-08"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Physics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Extended Phase Space"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2024 Pin-Chun Pai"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/125667"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-08-01","The student, Pin-Chun Pai, accepted the attached license on 2024-06-13 at 10:23.","The student, Pin-Chun Pai, submitted this Dissertation for approval on 2024-06-13 at 10:33.","This Dissertation was approved for publication on 2024-06-18 at 16:46.","DSpace SAF Submission Ingestion Package generated from Vireo submission #20831 on 2025-02-04 at 21:15:41","In this thesis we will explore the embedding degree of freedom, corner symmetries, and the associated charges for general gauge systems with spacetime boundaries. We begin the discussion by focusing on the extended phase space in chapters 1 and 2. These chapters are heavily based on the papers [1] and [2], respectively. In the introduction of Chapter 1, we take the opportunity to review the symmetries of gauge theories and describe how these symmetries are related to the conserved charges in the covariant phase space formalism. We recall the issue of integrability in theories with diffeomorphism symmetry, and then propose the extended phase space approach to deal with it. In particular, the phase space of diffeomorphism-invariant theories should be extended to account for embeddings associated with the boundary of subregions. We do so by emphasizing the importance of careful treatment of embeddings in all aspects of the covariant phase space formalism. In doing so, we introduce a new notion of extending the field space associated with embeddings. This extension has the important feature that the Noether charges associated with all extended corner symmetries are, in fact, integrable, but not necessarily conserved. We then demonstrate that the charges provide a representation of the extended corner symmetry via the Poisson bracket, without requiring a central extension. In Chapter 2, we continue in the theme of extended phase space. Here we generalize the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs (φ, Φ), where φ is a Lie algebroid morphism and Φ is a field configuration in the non-extended sense. With these data, we outline a powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons theory and demonstrate its utility by recalculating the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories in the appendix. Finally, in Chapter 3, we shift our focus to complexity, another important quantity associated to quantum gravity theory. The content of this chapter is drawn from the work of [3]. Motivated by holographic proposal, we study notions of complexity for link complement states in Chern-Simons theory with compact gauge group G. These states are obtained via the Euclidean path integral on the complement of n-component links within a 3-manifold M3. For the Abelian theory at level k, we discover that a natural set of fundamental gates exists and allows us to identify complexity as differences of linking numbers modulo k. Such linking numbers can be viewed as coordinates which embed all link complement states into Z⊗n(n−1)/2 k and the complexity is identified as the distance with respect to a particular norm. For non-Abelian Chern-Simons theories, the situation is much more complicated. We focus here on torus link states and show that the problem can be reduced to defining complexity for a single knot complement state. We propose a systematic approach to select a set of minimal universal generators for single knot complement states and then evaluate complexity using such generators. We provide a detailed illustration for SU(2)k Chern-Simons theory and these results can be extended to a general compact gauge group."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Embedding degree of freedom, integrable charges and extended phase space"]}]}],"canonical_facts":{"dc:contributor":["Leigh, Robert","Faulkner, Thomas","Fradkin, Eduardo","Hughes, Taylor"],"dc:creator":["Pai, Pin-Chun"],"dc:date":["2024-06-18","2024-08"],"dc:description":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2026-08-01","The student, Pin-Chun Pai, accepted the attached license on 2024-06-13 at 10:23.","The student, Pin-Chun Pai, submitted this Dissertation for approval on 2024-06-13 at 10:33.","This Dissertation was approved for publication on 2024-06-18 at 16:46.","DSpace SAF Submission Ingestion Package generated from Vireo submission #20831 on 2025-02-04 at 21:15:41","In this thesis we will explore the embedding degree of freedom, corner symmetries, and the associated charges for general gauge systems with spacetime boundaries. We begin the discussion by focusing on the extended phase space in chapters 1 and 2. These chapters are heavily based on the papers [1] and [2], respectively. In the introduction of Chapter 1, we take the opportunity to review the symmetries of gauge theories and describe how these symmetries are related to the conserved charges in the covariant phase space formalism. We recall the issue of integrability in theories with diffeomorphism symmetry, and then propose the extended phase space approach to deal with it. In particular, the phase space of diffeomorphism-invariant theories should be extended to account for embeddings associated with the boundary of subregions. We do so by emphasizing the importance of careful treatment of embeddings in all aspects of the covariant phase space formalism. In doing so, we introduce a new notion of extending the field space associated with embeddings. This extension has the important feature that the Noether charges associated with all extended corner symmetries are, in fact, integrable, but not necessarily conserved. We then demonstrate that the charges provide a representation of the extended corner symmetry via the Poisson bracket, without requiring a central extension. In Chapter 2, we continue in the theme of extended phase space. Here we generalize the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs (φ, Φ), where φ is a Lie algebroid morphism and Φ is a field configuration in the non-extended sense. With these data, we outline a powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons theory and demonstrate its utility by recalculating the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories in the appendix. Finally, in Chapter 3, we shift our focus to complexity, another important quantity associated to quantum gravity theory. The content of this chapter is drawn from the work of [3]. Motivated by holographic proposal, we study notions of complexity for link complement states in Chern-Simons theory with compact gauge group G. These states are obtained via the Euclidean path integral on the complement of n-component links within a 3-manifold M3. For the Abelian theory at level k, we discover that a natural set of fundamental gates exists and allows us to identify complexity as differences of linking numbers modulo k. Such linking numbers can be viewed as coordinates which embed all link complement states into Z⊗n(n−1)/2 k and the complexity is identified as the distance with respect to a particular norm. For non-Abelian Chern-Simons theories, the situation is much more complicated. We focus here on torus link states and show that the problem can be reduced to defining complexity for a single knot complement state. We propose a systematic approach to select a set of minimal universal generators for single knot complement states and then evaluate complexity using such generators. We provide a detailed illustration for SU(2)k Chern-Simons theory and these results can be extended to a general compact gauge group."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/125667"],"dc:language":["en","eng"],"dc:rights":["Copyright 2024 Pin-Chun Pai"],"dc:subject":["Extended Phase Space"],"dc:title":["Embedding degree of freedom, integrable charges and extended phase space"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Physics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:02Z"}