{"id":"https://openalex.org/W7220711510","doi":"https://doi.org/10.48550/arxiv.2610.04362","title":"Classification of commutation relation for multi-chordal SLE","display_name":"Classification of commutation relation for multi-chordal SLE","publication_year":2026,"publication_date":"2026-10-03","ids":{"openalex":"https://openalex.org/W7220711510","doi":"https://doi.org/10.48550/arxiv.2610.04362"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.04362","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04362","pdf_url":"https://arxiv.org/pdf/2610.04362","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/2610.04362","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5157720596","display_name":"Mo Chen","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Chen, Mo","raw_affiliation_strings":["Tsinghua University , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Tsinghua University , China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5126810958","display_name":"Chongzhi Huang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Huang, Chongzhi","raw_affiliation_strings":["Tsinghua University , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Tsinghua University , China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5157758695","display_name":"Hao Wu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wu, Hao","raw_affiliation_strings":["Tsinghua University , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Tsinghua University , China","institution_ids":["https://openalex.org/I99065089"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I99065089"],"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.6973000168800354,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.6973000168800354,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.14399999380111694,"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":0.01810000091791153,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/schramm-loewner-evolution","display_name":"Schramm-Loewner evolution","score":0.8500000238418579},{"id":"https://openalex.org/keywords/chordal-sle","display_name":"chordal SLE","score":0.7319999933242798},{"id":"https://openalex.org/keywords/commutation-relations","display_name":"commutation relations","score":0.7080000042915344},{"id":"https://openalex.org/keywords/conformal-ward-identities","display_name":"conformal Ward identities","score":0.9089999794960022},{"id":"https://openalex.org/keywords/translation-operators","display_name":"translation operators","score":0.5669999718666077},{"id":"https://openalex.org/keywords/knizhnik-zamolodchikov-equations","display_name":"Knizhnik-Zamolodchikov equations","score":0.8700000047683716},{"id":"https://openalex.org/keywords/uniform-spanning-tree","display_name":"uniform spanning tree","score":0.8360000252723694},{"id":"https://openalex.org/keywords/partition-function","display_name":"partition function","score":0.5619999766349792},{"id":"https://openalex.org/keywords/jordan-structure","display_name":"Jordan structure","score":0.33399999141693115},{"id":"https://openalex.org/keywords/scaling-exponents","display_name":"scaling exponents","score":0.5379999876022339}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8698999881744385},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.6033999919891357},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4034000039100647},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3991999924182892},{"id":"https://openalex.org/C33222762","wikidata":"https://www.wikidata.org/wiki/Q321119","display_name":"Identity function","level":2,"score":0.388700008392334},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.38370001316070557},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.37599998712539673},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.3425000011920929},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.34150001406669617},{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.33160001039505005},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.328900009393692},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.3093999922275543},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.3066999912261963},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.30309998989105225},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.29980000853538513},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.2971000075340271},{"id":"https://openalex.org/C49766605","wikidata":"https://www.wikidata.org/wiki/Q207643","display_name":"Linear map","level":2,"score":0.29649999737739563},{"id":"https://openalex.org/C152401794","wikidata":"https://www.wikidata.org/wiki/Q583760","display_name":"Antisymmetric relation","level":2,"score":0.2870999872684479},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.2854999899864197},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.2808000147342682},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.2782999873161316},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.2718999981880188},{"id":"https://openalex.org/C200741047","wikidata":"https://www.wikidata.org/wiki/Q1957758","display_name":"Integrable system","level":2,"score":0.2694999873638153},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.26820001006126404},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.26750001311302185},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.2653999924659729},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.26510000228881836},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.26350000500679016},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.26019999384880066},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.25450000166893005},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.25360000133514404},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.250900000333786},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.25029999017715454}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.04362","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04362","pdf_url":"https://arxiv.org/pdf/2610.04362","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.2610.04362","doi":"https://doi.org/10.48550/arxiv.2610.04362","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.04362","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:2610.04362","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04362","pdf_url":"https://arxiv.org/pdf/2610.04362","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":[{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220711510.pdf","grobid_xml":"https://content.openalex.org/works/W7220711510.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W3048652609","https://openalex.org/W4388684972","https://openalex.org/W4396913330","https://openalex.org/W4399554454","https://openalex.org/W4416557093","https://openalex.org/W7165644246","https://openalex.org/W7213589091"],"related_works":["https://openalex.org/W7213589091","https://openalex.org/W4247052474","https://openalex.org/W4416557093","https://openalex.org/W4240177850","https://openalex.org/W3106202260","https://openalex.org/W4415314064","https://openalex.org/W4404340899","https://openalex.org/W6958912487","https://openalex.org/W2039050839","https://openalex.org/W6891903463"],"abstract_inverted_index":{"We":[0],"classify":[1],"the":[2,7,29,51,57,72,75,98,104,122,130,135,144,156,167,179],"linear":[3,108],"solution":[4,78,170],"spaces":[5],"of":[6,56,74,83,90,178,197,211],"chordal":[8],"Belavin--Polyakov--Zamolodchikov":[9],"(BPZ)":[10],"equations,":[11],"which":[12],"arise":[13],"in":[14],"Dub\u00e9dat's":[15],"commutation":[16],"relations":[17],"for":[18,208],"multiple":[19],"Schramm--Loewner":[20],"evolutions":[21],"(SLE).":[22],"Without":[23],"imposing":[24],"growth":[25],"conditions,":[26],"we":[27,49,64,192],"determine":[28,50,66],"subspaces":[30],"selected":[31],"by":[32,125,150],"successive":[33],"conformal":[34],"Ward":[35,99,132,158],"identities":[36],"and":[37,44,53,62,71,102,114,121,134,152,190,201],"their":[38],"exact":[39],"dimensions.":[40],"For":[41,110],"every":[42],"$\u03ba>0$":[43],"BPZ":[45,117,199],"spectral":[46],"parameter":[47],"$\u03bb\\in\\mathbb{R}$,":[48],"spectrum":[52],"Jordan":[54],"structure":[55],"translation":[58],"operator.":[59],"When":[60],"$\u03bb=0$":[61,120],"$\u03ba\\in(0,8)$,":[63,115],"further":[65],"all":[67],"admissible":[68],"scaling":[69],"exponents":[70],"dimensions":[73],"corresponding":[76],"translation-invariant":[77,116],"spaces.":[79],"A":[80],"triangular":[81],"change":[82],"derivative":[84],"coordinates":[85],"yields":[86],"a":[87,212],"first-order":[88],"system":[89],"rational":[91],"Knizhnik--Zamolodchikov":[92],"type,":[93],"allowing":[94],"us":[95],"to":[96,106],"represent":[97],"operators":[100],"explicitly":[101],"reduce":[103],"classification":[105],"finite-dimensional":[107],"algebra.":[109],"$2N$":[111],"boundary":[112],"points":[113],"solutions":[118,200],"with":[119,216],"homogeneity":[123],"required":[124],"M\u00f6bius":[126],"covariance":[127],"automatically":[128],"satisfy":[129],"third":[131,157],"identity":[133,159],"usual":[136],"power-law":[137],"bound.":[138],"This":[139],"recovers,":[140],"under":[141],"weaker":[142],"assumptions,":[143],"Catalan":[145,174],"dimension":[146],"formula":[147],"previously":[148],"established":[149],"Flores":[151],"Kleban.":[153],"At":[154],"$\u03ba=8$,":[155],"imposes":[160],"an":[161,194],"additional":[162],"constraint":[163],"when":[164],"$N\\ge2$,":[165],"but":[166],"full":[168],"M\u00f6bius-covariant":[169],"space":[171],"still":[172],"has":[173],"dimension,":[175],"establishing":[176],"completeness":[177],"partition":[180,206],"functions":[181,207],"constructed":[182],"from":[183],"uniform":[184],"spanning":[185],"trees.":[186],"Finally,":[187],"at":[188],"$\u03ba=4$":[189],"$\u03bb>0$,":[191],"construct":[193],"explicit":[195],"basis":[196],"positive":[198],"identify":[202],"its":[203],"elements":[204],"as":[205],"level":[209],"lines":[210],"Gaussian":[213],"free":[214],"field":[215],"suitable":[217],"harmonic":[218],"means.":[219]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}