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UID:78ed3afbe49b3e4b713d799aaf8a7d28
CATEGORIES:Lie Group Quantum Mathematics Seminar
CREATED:20230410T101642
SUMMARY:Modular invariance of logarithmic intertwining operators
LOCATION:Hill 705
DESCRIPTION:Abstract I will discuss a proof of the conjecture of almost twenty years on
  the modular invariance of (logarithmic) intertwining operators.Let V be a 
 C_2-cofinite vertex operator algebra of positive energy.The conjecture stat
 es that the vector space spanned by pseudo-q-tracesshifted by -c/24 of prod
 ucts of (logarithmic) intertwining operators among grading-restricted gener
 alized V-modules is a module forthe modular group SL(2, Z). In 2015, Fiorda
 lisi proved that such pseudo-q-traces are absolutely convergent and have th
 e genus-oneassociativity property and some other properties. Recently, I ha
 ve proved this conjecture completely. This modular invariace givesa constru
 ction of C_2-cofinite genus-one logarithmic conformal field theories. It mi
 ght play an important role in the future proofs of theconjectures on the co
 rresponding braided tensor categories, especially their rigidity and modula
 rity.\n
X-ALT-DESC;FMTTYPE=text/html:<p><strong style="color: #242424; font-family: 'Segoe UI', 'Segoe UI Web (W
 est European)', 'Segoe UI', -apple-system, BlinkMacSystemFont, Roboto, 'Hel
 vetica Neue', sans-serif; font-size: 15px; letter-spacing: normal; orphans:
  2; text-align: start; text-indent: 0px; text-transform: none; white-space:
  normal; widows: 2; word-spacing: 0px; background-color: #ffffff;">Abstract
 &nbsp;</strong>I will discuss a proof of the conjecture of almost twenty ye
 ars&nbsp;on the modular invariance of (logarithmic) intertwining operators.
 <br style="color: #242424; font-family: 'Segoe UI', 'Segoe UI Web (West Eur
 opean)', 'Segoe UI', -apple-system, BlinkMacSystemFont, Roboto, 'Helvetica 
 Neue', sans-serif; font-size: 15px; font-weight: 400; letter-spacing: norma
 l; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; w
 hite-space: normal; widows: 2; word-spacing: 0px; background-color: #ffffff
 ;" aria-hidden="true">Let V be a C_2-cofinite vertex operator algebra of po
 sitive energy.The conjecture states that the vector space spanned by pseudo
 -q-traces<br style="color: #242424; font-family: 'Segoe UI', 'Segoe UI Web 
 (West European)', 'Segoe UI', -apple-system, BlinkMacSystemFont, Roboto, 'H
 elvetica Neue', sans-serif; font-size: 15px; font-weight: 400; letter-spaci
 ng: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform
 : none; white-space: normal; widows: 2; word-spacing: 0px; background-color
 : #ffffff;" aria-hidden="true">shifted by -c/24 of products of (logarithmic
 ) intertwining operators&nbsp;among grading-restricted generalized V-module
 s is a module for<br style="color: #242424; font-family: 'Segoe UI', 'Segoe
  UI Web (West European)', 'Segoe UI', -apple-system, BlinkMacSystemFont, Ro
 boto, 'Helvetica Neue', sans-serif; font-size: 15px; font-weight: 400; lett
 er-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-t
 ransform: none; white-space: normal; widows: 2; word-spacing: 0px; backgrou
 nd-color: #ffffff;" aria-hidden="true">the modular group SL(2, Z). In 2015,
  Fiordalisi proved that such&nbsp;pseudo-q-traces are absolutely convergent
  and have the genus-one<br style="color: #242424; font-family: 'Segoe UI', 
 'Segoe UI Web (West European)', 'Segoe UI', -apple-system, BlinkMacSystemFo
 nt, Roboto, 'Helvetica Neue', sans-serif; font-size: 15px; font-weight: 400
 ; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; 
 text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; ba
 ckground-color: #ffffff;" aria-hidden="true">associativity property and som
 e other properties. Recently, I have&nbsp;proved this conjecture completely
 . This modular invariace gives<br style="color: #242424; font-family: 'Sego
 e UI', 'Segoe UI Web (West European)', 'Segoe UI', -apple-system, BlinkMacS
 ystemFont, Roboto, 'Helvetica Neue', sans-serif; font-size: 15px; font-weig
 ht: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent
 : 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 
 0px; background-color: #ffffff;" aria-hidden="true">a construction of C_2-c
 ofinite genus-one logarithmic conformal field&nbsp;theories. It might play 
 an important role in the future proofs of the<br style="color: #242424; fon
 t-family: 'Segoe UI', 'Segoe UI Web (West European)', 'Segoe UI', -apple-sy
 stem, BlinkMacSystemFont, Roboto, 'Helvetica Neue', sans-serif; font-size: 
 15px; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: sta
 rt; text-indent: 0px; text-transform: none; white-space: normal; widows: 2;
  word-spacing: 0px; background-color: #ffffff;" aria-hidden="true">conjectu
 res on the corresponding braided tensor categories, especially&nbsp;their r
 igidity and modularity.</p>
CONTACT:  Speaker Yi-Zhi Huang, Rutgers University
DTSTAMP:20260827T154451
DTSTART;TZID=America/New_York:20230428T121000
DTEND;TZID=America/New_York:20230428T131000
SEQUENCE:0
TRANSP:OPAQUE
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