{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/73434"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/73434","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Quantum geometric Langlands correspondence in positive characteristic: the GLN case","abstract":"Let C be a smooth connected projective curve of genus > 1 over an algebraically closed field k of characteristic p > 0, and c [epsilon] k \\ Fp. Let BunN be the stack of rank N vector bundles on C and Ldet the line bundle on BunN given by determinant of derived global sections. In this thesis, we construct an equivalence of derived categories of modules for certain localizations of the twisted crystalline differential operator algebras DBunNet and DBunN, L-1/cdet The first step of the argument is the same as that of [BB] for the non-quantum case: based on the Azumaya property of crystalline differential operators, the equivalence is constructed as a twisted version of Fourier-Mukai transform on the Hitchin fibration. However, there are some new ingredients. Along the way we introduce a generalization of p-curvature for line bundles with non-flat connections, and construct a Liouville vector field on the space of de Rham local systems on C.","abstract_html":"Let C be a smooth connected projective curve of genus > 1 over an algebraically closed field k of characteristic p > 0, and c [epsilon] k \\ Fp. Let BunN be the stack of rank N vector bundles on C and Ldet the line bundle on BunN given by determinant of derived global sections. In this thesis, we construct an equivalence of derived categories of modules for certain localizations of the twisted crystalline differential operator algebras DBunNet and DBunN, L-1/cdet The first step of the argument is the same as that of [BB] for the non-quantum case: based on the Azumaya property of crystalline differential operators, the equivalence is constructed as a twisted version of Fourier-Mukai transform on the Hitchin fibration. However, there are some new ingredients. Along the way we introduce a generalization of p-curvature for line bundles with non-flat connections, and construct a Liouville vector field on the space of de Rham local systems on C.","abstract_has_math":false,"creators":["Travkin, Roman (Roman Mikhailovich)"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["Roman Bezrukavnikov."],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012","date_published":"2012","updated_at":"2026-07-22T22:21:55Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/73434","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Roman Bezrukavnikov."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:creator","label":"Author","values":["Travkin, Roman (Roman Mikhailovich)"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2012-09-27T18:11:41Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2012-09-27T18:11:41Z"]},{"key":"dc:date.issued","label":"Date","values":["2012"]},{"key":"dc:publisher","label":"Institution","values":["Massachusetts Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/73434"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.","In title on title page, \"N\" of GLN appears as subscript of upper case letter N. Cataloged from PDF version of thesis.","Includes bibliographical references (p. 73)."]},{"key":"dc:description.abstract","label":"Abstract","values":["Let C be a smooth connected projective curve of genus > 1 over an algebraically closed field k of characteristic p > 0, and c [epsilon] k \\ Fp. Let BunN be the stack of rank N vector bundles on C and Ldet the line bundle on BunN given by determinant of derived global sections. In this thesis, we construct an equivalence of derived categories of modules for certain localizations of the twisted crystalline differential operator algebras DBunNet and DBunN, L-1/cdet The first step of the argument is the same as that of [BB] for the non-quantum case: based on the Azumaya property of crystalline differential operators, the equivalence is constructed as a twisted version of Fourier-Mukai transform on the Hitchin fibration. However, there are some new ingredients. Along the way we introduce a generalization of p-curvature for line bundles with non-flat connections, and construct a Liouville vector field on the space of de Rham local systems on C."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Quantum geometric Langlands correspondence in positive characteristic: the GLN case"]}]}],"canonical_facts":{"dc:contributor.advisor":["Roman Bezrukavnikov."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:creator":["Travkin, Roman (Roman Mikhailovich)"],"dc:date.accessioned":["2012-09-27T18:11:41Z"],"dc:date.available":["2012-09-27T18:11:41Z"],"dc:date.issued":["2012"],"dc:description":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.","In title on title page, \"N\" of GLN appears as subscript of upper case letter N. Cataloged from PDF version of thesis.","Includes bibliographical references (p. 73)."],"dc:description.abstract":["Let C be a smooth connected projective curve of genus > 1 over an algebraically closed field k of characteristic p > 0, and c [epsilon] k \\ Fp. Let BunN be the stack of rank N vector bundles on C and Ldet the line bundle on BunN given by determinant of derived global sections. In this thesis, we construct an equivalence of derived categories of modules for certain localizations of the twisted crystalline differential operator algebras DBunNet and DBunN, L-1/cdet The first step of the argument is the same as that of [BB] for the non-quantum case: based on the Azumaya property of crystalline differential operators, the equivalence is constructed as a twisted version of Fourier-Mukai transform on the Hitchin fibration. However, there are some new ingredients. Along the way we introduce a generalization of p-curvature for line bundles with non-flat connections, and construct a Liouville vector field on the space of de Rham local systems on C."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/73434"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Quantum geometric Langlands correspondence in positive characteristic: the GLN case"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:21:55Z"}