{"id":"https://openalex.org/W7148945793","doi":"https://doi.org/10.48550/arxiv.2604.01275","title":"Descending into the Modular Bootstrap","display_name":"Descending into the Modular Bootstrap","publication_year":2026,"publication_date":"2026-04-01","ids":{"openalex":"https://openalex.org/W7148945793","doi":"https://doi.org/10.48550/arxiv.2604.01275"},"language":"en","primary_location":{"id":"pmh:oai:HAL:hal-05584933v1","is_oa":false,"landing_page_url":"https://hal.science/hal-05584933","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"2026","raw_type":"info:eu-repo/semantics/preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.01275","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5070838186","display_name":"Nathan Benjamin","orcid":"https://orcid.org/0000-0003-3661-6563"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Benjamin, Nathan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034046294","display_name":"A. Liam Fitzpatrick","orcid":"https://orcid.org/0000-0002-8818-2069"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fitzpatrick, A. Liam","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5132875492","display_name":"Wei Li","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Wei","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5051781797","display_name":"Jesse Thaler","orcid":"https://orcid.org/0000-0002-2406-8160"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Thaler, Jesse","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.2849000096321106,"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":0.2849000096321106,"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":0.1671999990940094,"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/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.10849999636411667,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/modular-invariance","display_name":"Modular invariance","score":0.7062000036239624},{"id":"https://openalex.org/keywords/modular-design","display_name":"Modular design","score":0.6672000288963318},{"id":"https://openalex.org/keywords/partition-function","display_name":"Partition function (quantum field theory)","score":0.5837000012397766},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.5102999806404114},{"id":"https://openalex.org/keywords/conformal-field-theory","display_name":"Conformal field theory","score":0.4708000123500824},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.45570001006126404},{"id":"https://openalex.org/keywords/conformal-map","display_name":"Conformal map","score":0.4262999892234802},{"id":"https://openalex.org/keywords/torus","display_name":"Torus","score":0.41530001163482666}],"concepts":[{"id":"https://openalex.org/C97900337","wikidata":"https://www.wikidata.org/wiki/Q60367","display_name":"Modular invariance","level":3,"score":0.7062000036239624},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.6672000288963318},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6090999841690063},{"id":"https://openalex.org/C2778401447","wikidata":"https://www.wikidata.org/wiki/Q7140637","display_name":"Partition function (quantum field theory)","level":2,"score":0.5837000012397766},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.5102999806404114},{"id":"https://openalex.org/C54613284","wikidata":"https://www.wikidata.org/wiki/Q1191271","display_name":"Conformal field theory","level":3,"score":0.4708000123500824},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.45570001006126404},{"id":"https://openalex.org/C98214594","wikidata":"https://www.wikidata.org/wiki/Q850275","display_name":"Conformal map","level":2,"score":0.4262999892234802},{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.41530001163482666},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4041000008583069},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3776000142097473},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.3725999891757965},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.34779998660087585},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.3366999924182892},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.33230000734329224},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.3237000107765198},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3091000020503998},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3084000051021576},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29109999537467957},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.27730000019073486},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.25949999690055847},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.25429999828338623},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2531999945640564},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.25049999356269836}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:HAL:hal-05584933v1","is_oa":false,"landing_page_url":"https://hal.science/hal-05584933","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"2026","raw_type":"info:eu-repo/semantics/preprint"},{"id":"doi:10.48550/arxiv.2604.01275","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.01275","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.01275","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.01275","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"In":[0],"this":[1,66],"paper,":[2],"we":[3,52,77],"attempt":[4],"to":[5,22,54,80,102,111],"explore":[6],"the":[7,23,41,56,63,104,109,112,119,136,140,189,197],"landscape":[8],"of":[9,34,43,121,139,161,175,200],"two-dimensional":[10],"conformal":[11],"field":[12],"theories":[13],"(2d":[14],"CFTs)":[15],"by":[16,40],"efficiently":[17],"searching":[18],"for":[19,183],"numerical":[20],"solutions":[21],"modular":[24,70,176],"bootstrap":[25,177],"equation":[26],"using":[27],"machine-learning-style":[28],"optimization.":[29],"The":[30,97,116],"torus":[31],"partition":[32,67,149],"function":[33,68],"a":[35,73,100,122,158,172,184],"2d":[36],"CFT":[37,148],"is":[38,69,99,118,128],"fixed":[39],"spectrum":[42,110],"its":[44,48],"primary":[45,83],"operators":[46],"and":[47,156,164],"chiral":[49],"algebra,":[50],"which":[51,76],"take":[53],"be":[55],"Virasoro":[57],"algebra":[58],"with":[59,107,151],"$c&gt;1$.":[60],"We":[61,143,179],"translate":[62],"requirement":[64],"that":[65,91,127,166],"invariant":[71],"into":[72],"loss":[74,141],"function,":[75],"then":[78],"minimize":[79],"identify":[81],"possible":[82],"spectra.":[84],"Our":[85],"approach":[86],"involves":[87],"two":[88],"technical":[89],"innovations":[90],"facilitate":[92],"finding":[93],"reliable":[94],"candidate":[95,146],"CFTs.":[96],"first":[98],"strategy":[101],"estimate":[103],"uncertainty":[105],"associated":[106],"truncating":[108],"lowest":[113],"dimension":[114],"operators.":[115],"second":[117],"use":[120],"new":[123],"singular-value-based":[124],"optimizer":[125],"(Sven)":[126],"more":[129,185],"effective":[130],"than":[131,196],"gradient":[132],"descent":[133],"at":[134],"navigating":[135],"hierarchical":[137],"structure":[138],"landscape.":[142],"numerically":[144],"construct":[145],"truncated":[147],"functions":[150],"central":[152],"charges":[153],"between":[154],"1":[155],"$\\frac{8}{7}$,":[157],"range":[159],"devoid":[160],"known":[162],"examples,":[163],"argue":[165],"these":[167],"candidates":[168],"likely":[169],"come":[170],"from":[171],"continuous":[173],"space":[174],"solutions.":[178],"also":[180],"provide":[181],"evidence":[182],"stringent":[186],"constraint":[187],"on":[188],"spectral":[190],"gap":[191],"near":[192],"$c":[193],"=":[194],"1$":[195],"existing":[198],"bound":[199],"$\u0394_{\\rm":[201],"gap}":[202],"\\le":[203],"\\frac{c}{6}":[204],"+":[205],"\\frac{1}{3}$.":[206]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-04-04T00:00:00"}
