{"id":"https://openalex.org/W7214071014","doi":"https://doi.org/10.48550/arxiv.2609.25215","title":"Bulk OPE Coefficients of the $E$-Series Virasoro Minimal Models","display_name":"Bulk OPE Coefficients of the $E$-Series Virasoro Minimal Models","publication_year":2026,"publication_date":"2026-09-21","ids":{"openalex":"https://openalex.org/W7214071014","doi":"https://doi.org/10.48550/arxiv.2609.25215"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.25215","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.25215","pdf_url":"https://arxiv.org/pdf/2609.25215","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.25215","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5152554622","display_name":"Amaury Lhoste","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I166644833","display_name":"Kavli Institute for the Physics and Mathematics of the Universe","ror":"https://ror.org/02chw6z69","country_code":"JP","type":"facility","lineage":["https://openalex.org/I166644833","https://openalex.org/I4405286593","https://openalex.org/I74801974"]},{"id":"https://openalex.org/I2746051580","display_name":"Universit\u00e9 Paris Sciences et Lettres","ror":"https://ror.org/013cjyk83","country_code":"FR","type":"education","lineage":["https://openalex.org/I2746051580"]},{"id":"https://openalex.org/I29607241","display_name":"\u00c9cole Normale Sup\u00e9rieure - PSL","ror":"https://ror.org/05a0dhs15","country_code":"FR","type":"other","lineage":["https://openalex.org/I2746051580","https://openalex.org/I29607241"]},{"id":"https://openalex.org/I74801974","display_name":"The University of Tokyo","ror":"https://ror.org/057zh3y96","country_code":"JP","type":"education","lineage":["https://openalex.org/I74801974"]}],"countries":["FR","JP"],"is_corresponding":false,"raw_author_name":"Lhoste, Amaury","raw_affiliation_strings":["Kavli Institute for the Physics and Mathematics of the Universe , UTIAS , University of Tokyo , Kashiwa , Chiba 277-8583 , Japan","\u00c9cole normale sup\u00e9rieure , Universit\u00e9 PSL , 45 rue d'Ulm , 75005 Paris , France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Kavli Institute for the Physics and Mathematics of the Universe , UTIAS , University of Tokyo , Kashiwa , Chiba 277-8583 , Japan","institution_ids":["https://openalex.org/I166644833","https://openalex.org/I74801974"]},{"raw_affiliation_string":"\u00c9cole normale sup\u00e9rieure , Universit\u00e9 PSL , 45 rue d'Ulm , 75005 Paris , France","institution_ids":["https://openalex.org/I2746051580","https://openalex.org/I29607241"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056137903","display_name":"Jiaxin Qiao","orcid":"https://orcid.org/0000-0001-9342-5662","observed_orcids":["https://orcid.org/0000-0001-9342-5662"]},"institutions":[{"id":"https://openalex.org/I166644833","display_name":"Kavli Institute for the Physics and Mathematics of the Universe","ror":"https://ror.org/02chw6z69","country_code":"JP","type":"facility","lineage":["https://openalex.org/I166644833","https://openalex.org/I4405286593","https://openalex.org/I74801974"]},{"id":"https://openalex.org/I74801974","display_name":"The University of Tokyo","ror":"https://ror.org/057zh3y96","country_code":"JP","type":"education","lineage":["https://openalex.org/I74801974"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Qiao, Jiaxin","raw_affiliation_strings":["Center for Data-Driven Discovery (CD3) , Kavli IPMU , University of Tokyo , Chiba 277-8583 , Japan","Kavli Institute for the Physics and Mathematics of the Universe , UTIAS , University of Tokyo , Kashiwa , Chiba 277-8583 , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Data-Driven Discovery (CD3) , Kavli IPMU , University of Tokyo , Chiba 277-8583 , Japan","institution_ids":["https://openalex.org/I166644833","https://openalex.org/I74801974"]},{"raw_affiliation_string":"Kavli Institute for the Physics and Mathematics of the Universe , UTIAS , University of Tokyo , Kashiwa , Chiba 277-8583 , Japan","institution_ids":["https://openalex.org/I166644833","https://openalex.org/I74801974"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5152508059","display_name":"Masahito Yamazaki","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yamazaki, Masahito","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9735999703407288,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9735999703407288,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10025","display_name":"Black Holes and Theoretical Physics","score":0.02239999920129776,"subfield":{"id":"https://openalex.org/subfields/3106","display_name":"Nuclear and High Energy Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.00019999999494757503,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/virasoro-minimal-models","display_name":"Virasoro minimal models","score":0.8960000276565552},{"id":"https://openalex.org/keywords/conformal-bootstrap","display_name":"conformal bootstrap","score":0.7850000262260437},{"id":"https://openalex.org/keywords/6j-symbols","display_name":"6j symbols","score":0.9089999794960022},{"id":"https://openalex.org/keywords/central-charge","display_name":"central charge","score":0.6119999885559082},{"id":"https://openalex.org/keywords/crossing-equations","display_name":"crossing equations","score":0.7760000228881836},{"id":"https://openalex.org/keywords/bulk-locality","display_name":"bulk locality","score":0.8659999966621399},{"id":"https://openalex.org/keywords/exceptional-lie-algebras","display_name":"exceptional Lie algebras","score":0.6890000104904175}],"concepts":[{"id":"https://openalex.org/C2778681612","wikidata":"https://www.wikidata.org/wiki/Q15427593","display_name":"Minimal models","level":2,"score":0.6815000176429749},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.6757000088691711},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.6699000000953674},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5523999929428101},{"id":"https://openalex.org/C98214594","wikidata":"https://www.wikidata.org/wiki/Q850275","display_name":"Conformal map","level":2,"score":0.5447999835014343},{"id":"https://openalex.org/C2777345500","wikidata":"https://www.wikidata.org/wiki/Q3658596","display_name":"Central charge","level":3,"score":0.5432999730110168},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5414999723434448},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5220000147819519},{"id":"https://openalex.org/C18199665","wikidata":"https://www.wikidata.org/wiki/Q3883909","display_name":"Operator product expansion","level":2,"score":0.4984000027179718},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.42579999566078186},{"id":"https://openalex.org/C2780032228","wikidata":"https://www.wikidata.org/wiki/Q6865324","display_name":"Minimal model","level":2,"score":0.3813999891281128},{"id":"https://openalex.org/C2779808786","wikidata":"https://www.wikidata.org/wiki/Q6664603","display_name":"Locality","level":2,"score":0.3720000088214874},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.36489999294281006},{"id":"https://openalex.org/C198252154","wikidata":"https://www.wikidata.org/wiki/Q1366201","display_name":"Virasoro algebra","level":5,"score":0.3452000021934509},{"id":"https://openalex.org/C159423971","wikidata":"https://www.wikidata.org/wiki/Q177251","display_name":"Associative property","level":2,"score":0.3319999873638153},{"id":"https://openalex.org/C130295795","wikidata":"https://www.wikidata.org/wiki/Q3550684","display_name":"Unitarity","level":2,"score":0.29409998655319214},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.2930999994277954},{"id":"https://openalex.org/C97900337","wikidata":"https://www.wikidata.org/wiki/Q60367","display_name":"Modular invariance","level":3,"score":0.2879999876022339},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.28519999980926514},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2851000130176544},{"id":"https://openalex.org/C143782950","wikidata":"https://www.wikidata.org/wiki/Q40460","display_name":"Orbifold","level":2,"score":0.26919999718666077},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2549000084400177}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.25215","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.25215","pdf_url":"https://arxiv.org/pdf/2609.25215","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.25215","doi":"https://doi.org/10.48550/arxiv.2609.25215","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.25215","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.25215","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.25215","pdf_url":"https://arxiv.org/pdf/2609.25215","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214071014.pdf","grobid_xml":"https://content.openalex.org/works/W7214071014.grobid-xml"},"referenced_works_count":77,"referenced_works":["https://openalex.org/W1584463974","https://openalex.org/W1966183300","https://openalex.org/W1970192247","https://openalex.org/W1974934079","https://openalex.org/W1976093463","https://openalex.org/W1977656703","https://openalex.org/W1979804473","https://openalex.org/W1984156733","https://openalex.org/W1993519201","https://openalex.org/W1993919670","https://openalex.org/W1994966164","https://openalex.org/W1998591827","https://openalex.org/W2002596439","https://openalex.org/W2004391251","https://openalex.org/W2008449309","https://openalex.org/W2009930506","https://openalex.org/W2010469783","https://openalex.org/W2013090014","https://openalex.org/W2014431958","https://openalex.org/W2016148523","https://openalex.org/W2022433426","https://openalex.org/W2022539570","https://openalex.org/W2031117500","https://openalex.org/W2032250899","https://openalex.org/W2034727892","https://openalex.org/W2037394029","https://openalex.org/W2037673637","https://openalex.org/W2039630417","https://openalex.org/W2043210214","https://openalex.org/W2052954763","https://openalex.org/W2054740422","https://openalex.org/W2058193907","https://openalex.org/W2058757923","https://openalex.org/W2063533767","https://openalex.org/W2065247876","https://openalex.org/W2065450544","https://openalex.org/W2068382363","https://openalex.org/W2069182458","https://openalex.org/W2074991751","https://openalex.org/W2076906094","https://openalex.org/W2078332443","https://openalex.org/W2080714821","https://openalex.org/W2081227483","https://openalex.org/W2082517833","https://openalex.org/W2082648758","https://openalex.org/W2085134639","https://openalex.org/W2087880934","https://openalex.org/W2087894886","https://openalex.org/W2090079307","https://openalex.org/W2093006813","https://openalex.org/W2100811836","https://openalex.org/W2101747874","https://openalex.org/W2119836827","https://openalex.org/W2127919256","https://openalex.org/W2135486303","https://openalex.org/W2136016367","https://openalex.org/W2149073224","https://openalex.org/W2154232758","https://openalex.org/W2161091926","https://openalex.org/W2166774662","https://openalex.org/W2167976557","https://openalex.org/W2209276742","https://openalex.org/W2786370293","https://openalex.org/W2915215992","https://openalex.org/W2949433516","https://openalex.org/W2952812443","https://openalex.org/W2997273956","https://openalex.org/W3031722725","https://openalex.org/W3170735844","https://openalex.org/W3180229875","https://openalex.org/W4206688401","https://openalex.org/W4362554919","https://openalex.org/W4404308589","https://openalex.org/W4407806693","https://openalex.org/W4408938302","https://openalex.org/W4414494101","https://openalex.org/W4416941944"],"related_works":["https://openalex.org/W2474001321","https://openalex.org/W2930064453","https://openalex.org/W4403884184","https://openalex.org/W2034727892","https://openalex.org/W1965165378","https://openalex.org/W4407806693","https://openalex.org/W2010221713","https://openalex.org/W2003259635","https://openalex.org/W3042987446","https://openalex.org/W1984156733"],"abstract_inverted_index":{"We":[0,98,148],"present":[1],"bulk-primary":[2],"operator-product":[3],"coefficients":[4,27,50,141],"for":[5,91,105,112,142,155],"every":[6],"Virasoro":[7,165],"minimal":[8],"model":[9],"with":[10],"an":[11],"$E_6$,":[12,143],"$E_7$,":[13,144],"or":[14],"$E_8$":[15],"modular":[16],"invariant.":[17],"Dividing":[18],"by":[19,39],"a":[20,34],"universal":[21],"product":[22],"of":[23,47,55],"chiral":[24],"generalized-minimal-model":[25],"OPE":[26,49,140,172],"reduces":[28,68],"the":[29,48,56,65,109,117,124],"conformal":[30],"bootstrap":[31],"problem":[32,37],"to":[33,69],"finite":[35,78,125],"algebraic":[36,83],"governed":[38],"root-of-unity":[40],"quantum-group":[41],"$6j$":[42],"symbols.":[43],"The":[44,130],"exceptional":[45],"factors":[46],"can":[51],"be":[52],"chosen":[53],"independent":[54],"second":[57],"Kac":[58,114],"indices":[59],"$s$,":[60],"and":[61,74,87,93,103,121,136,145,152,158,171],"their":[62],"dependence":[63],"on":[64],"central":[66],"charge":[67],"finitely":[70],"many":[71],"reference":[72],"cases":[73],"explicit":[75],"phases.":[76],"A":[77],"computer-assisted":[79],"calculation":[80,162],"yields":[81],"candidate":[82],"$s=1$":[84],"seed":[85,126],"data":[86],"high-precision":[88],"numerical":[89],"evidence":[90],"existence":[92,102],"uniqueness":[94,104],"modulo":[95],"primary-field":[96],"signs.":[97],"prove":[99],"that":[100],"exact":[101],"these":[106],"seeds":[107],"imply":[108],"corresponding":[110],"result":[111],"all":[113],"labels":[115],"within":[116],"genus-zero":[118],"bulk":[119,167],"locality":[120,168],"crossing":[122],"equations;":[123],"step":[127],"remains":[128],"uncertified.":[129],"three":[131],"families":[132],"require":[133],"$18$,":[134],"$32$,":[135],"$107$":[137],"representative":[138],"reduced":[139],"$E_8$,":[146],"respectively.":[147],"provide":[149],"reconstruction":[150],"conventions":[151],"coefficient":[153],"tables":[154],"both":[156],"unitary":[157],"nonunitary":[159],"models.":[160],"Our":[161],"assumes":[163],"only":[164],"symmetry,":[166],"(permutation":[169],"symmetry)":[170],"associativity":[173],"(crossing":[174],"equations).":[175]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-24T00:00:00"}