Geometric methods in vertex algebras

2016 Fiscal Year Annual Research Report

• Project/Area Number: 26610006
• Research Institution: Kyoto University
• Principal Investigator: 荒川 知幸 京都大学, 数理解析研究所, 准教授 (40377974)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 群論(含群の表現論） / 頂点作用素代数 / 随伴多様体 / 冪零軌道

Outline of Annual Research Achievements

研究代表者が導入した頂点代数の随伴多様体は、頂点代数の重要な幾何学的不変量であり、有限次元半単純リー環の普遍包絡環の原始イデアルの随伴多様体のアナロジーである。

頂点代数の随伴多様体はアフィンポアソン代数多様体として定義される。随伴多様体が点になるとき、頂点代数は平滑(lisse)と呼ばれ最も良い性質を持つことが知られているが、必ずしも平滑でなくても興味深い頂点代数が多数存在する。その典型的な例が許容アフィン頂点代数であり、対応する共型場理論は数学者・物理学者双方から注目されている。一方、最近のRastelli等による４次元-２次元対応により、許容アフィン頂点代数以外にも、同様な性質をもつ頂点代数の存在が明らかになってきた。これに関して、我々は許容アフィン頂点代数を拡張する概念として擬平滑頂点代数を導入し、さらに擬平滑頂点代数の指標がモジューラ微分方程式を満たし、従ってある種のモジューラ不変性を持つことを示した(川節和哉との共同研究、論文投稿中）。これは、４次元のN=2超対称性共型場理論から期待され、しかも物理学者が示し得なかった結果である。また、頂点代数の随伴多様体は、普遍包絡環の原始イデアルの随伴多様体場合とは異なり、必ずしも既約とはならないことが、擬平滑頂点代数の場合にはその随伴多様体が既約であることを予想し、これをW代数の例で確かめた（Anne Moreauとの共同研究、論文投稿中)。

これらの結果についてプリメータ研究所（カナダ)で行われた物理の研究集会で招待講演を行い、好評を得た。また多数の国外研究集会、セミナー、談話会でも招待講演を行った。

Research Products

• [Int'l Joint Research] University of Virginia/University of Denver(米国)
  • Country Name: U.S.A.
  • Counterpart Institution: University of Virginia/University of Denver
• [Int'l Joint Research] The University of Manchester(英国)
  • Country Name: UNITED KINGDOM
  • Counterpart Institution: The University of Manchester
• [Int'l Joint Research] the University of Poitiers(フランス)
  • Country Name: FRANCE
  • Counterpart Institution: the University of Poitiers
• [Int'l Joint Research] Universidade de Sao Paulo/Universidade Federal do ABC/Universidade Federal Fluminense(ブラジル)
  • Country Name: BRAZIL
  • Counterpart Institution: Universidade de Sao Paulo/Universidade Federal do ABC/Universidade Federal Fluminense
• [Int'l Joint Research] Academia Sinica(台湾)
  • Country Name: その他の国・地域
  • Counterpart Institution: Academia Sinica
• [Int'l Joint Research]
  • # of Other Countries: 2
• [Journal Article] Explicit generators in rectangular affine W-algebras of type A (2017)
  • Author(s): Tomoyuki Arakawa and Alexander Molev
  • Journal Title: Lett. Math. Phys. Volume: 107(1) Pages: 47-59
  • DOI: 10.1007/s11005-016-0890-2
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Vertex operator algebras associated with Z/kZ-codes (2017)
  • Author(s): Tomoyuki Arakawa, Hiromichi Yamada and Hiroshi Yamauchi
  • Journal Title: Springer Proceedings in Mathematics & Statistics Volume: 191 Pages: 513-521
  • DOI: 10.1007/978-981-10-2636-2_38
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Joseph ideals and lisse minimal W-algebras (2017)
  • Author(s): Tomoyuki Arakawa and Anne Moreau
  • Journal Title: J. Inst. Math. Jussieu Volume: 印刷中 Pages: -
  • DOI: 10.1017/S1474748016000025
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Weight representations of admissible affine vertex algebras (2017)
  • Author(s): Tomoyuki Arakawa, Vyacheslav Futorny and Luis Enrique Ramirez
  • Journal Title: Comm. Math. Phys. Volume: 印刷中 Pages: -
  • DOI: 10.1007/s00220-017-2872-3
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Orbifolds and cosets of minimal W-algebras (2017)
  • Author(s): Tomoyuki Arakawa, Thomas Creutzig, Kazuya Kawasetsu and Andrew Linshaw
  • Journal Title: Comm. Math. Phys. Volume: 印刷中 Pages: -
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Modular affine vertex algebras and baby Wakimoto modules (2016)
  • Author(s): Tomoyuki Arakawa and Weiqiang Wang
  • Journal Title: Proc. Symp. Pure Math. Volume: 92 Pages: 14-29
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Coset Construction of W-algebras (2017)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Algebra and Number Theory Seminar
  • Place of Presentation: UC Santa Cruz, USA
  • Year and Date: 2017-03-10 – 2017-03-10
  • Int'l Joint Research / Invited
• [Presentation] Vertex operator algebras and symplectic varieties (2017)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Colloquium
  • Place of Presentation: UC Santa Cruz, USA
  • Year and Date: 2017-03-07 – 2017-03-07
  • Int'l Joint Research / Invited
• [Presentation] GKO construction of minimal series W-algebras (2017)
  • Author(s): Tomoyuki Arakawa
  • Organizer: MIT Infinite Dimensional Algebra Seminar
  • Place of Presentation: MIT, USA
  • Year and Date: 2017-02-17 – 2017-02-17
  • Int'l Joint Research / Invited
• [Presentation] Moore-Tachikawa conjecture and chiral algebras of class S (2017)
  • Author(s): Tomoyuki Arakawa
  • Organizer: 7th Seminar on Conformal Field Theory
  • Place of Presentation: Technische Universitat Darmstadt, Germany
  • Year and Date: 2017-02-03 – 2017-02-03
  • Int'l Joint Research / Invited
• [Presentation] Moore-Tachikawa conjecture and chiral algebras of class S (2017)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Seminaire Geometrie Algebrique
  • Place of Presentation: Laboratoire Paul Painleve, Universite de Lille, France
  • Year and Date: 2017-01-31 – 2017-01-31
  • Int'l Joint Research / Invited
• [Presentation] Associated varieties of vertex algebras (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Exact operator algebras in superconformal field theories
  • Place of Presentation: Perimeter Institute for Theoretical Physics, Canada
  • Year and Date: 2016-12-14 – 2016-12-16
  • Int'l Joint Research / Invited
• [Presentation] Construction of genus zero chiral algebras of class S (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Exact operator algebras in superconformal field theories
  • Place of Presentation: Perimeter Institute for Theoretical Physics, Canada
  • Year and Date: 2016-12-14 – 2016-12-16
  • Int'l Joint Research / Invited
• [Presentation] Moore-Tachikawa conjecture and chiral algebras of class S (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: 60th Annual Meeting of the Australian Mathematical Society
  • Place of Presentation: Australian National University, Canberra, Australia
  • Year and Date: 2016-12-05 – 2016-12-05
  • Int'l Joint Research / Invited
• [Presentation] Moore-Tachikawa conjecture and chiral algebras of class S (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Sydney-UNSW Joint Colloquium
  • Place of Presentation: The University of Sydney, Australia
  • Year and Date: 2016-11-25 – 2016-11-25
  • Int'l Joint Research / Invited
• [Presentation] Vertex algebras and symplectic varieties (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Colloquium
  • Place of Presentation: the University of Poitiers, France
  • Year and Date: 2016-11-02 – 2016-11-02
  • Int'l Joint Research / Invited
• [Presentation] Vertex algebras and symplectic varieties (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Colloquium
  • Place of Presentation: University of Texas at Arlington
  • Year and Date: 2016-09-30 – 2016-09-30
  • Int'l Joint Research / Invited
• [Presentation] On the 2dTQFT associated with vertex algebras (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: MIT Infinite Dimensional Algebra Seminar
  • Place of Presentation: MIT, USA
  • Year and Date: 2016-09-09 – 2016-09-09
  • Int'l Joint Research / Invited
• [Presentation] On the 2d TQFT associated with vertex algebras (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Conference in Finite Groups and Vertex Algebras dedicated to Robert L. Griess on the occasion of his 71st birthday,
  • Place of Presentation: Academia Sinica, Taipei, Taiwan
  • Year and Date: 2016-08-26 – 2016-08-26
  • Int'l Joint Research / Invited
• [Presentation] Equivariant affine W-algebras (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Nilpotent Orbits and Representation Theory
  • Place of Presentation: Centro di Ricerca Matematica Ennio De Giorgi (CRM), Pisa, Italy
  • Year and Date: 2016-06-16 – 2016-06-16
  • Int'l Joint Research / Invited
• [Presentation] Beyond C2-cofinitneness (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Seminar on Algebra
  • Place of Presentation: University of Zagreb, Croatia
  • Year and Date: 2016-06-07 – 2016-06-07
  • Int'l Joint Research / Invited
• [Presentation] The Adamovic-Milas conjecture and application (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Colloquium
  • Place of Presentation: University of Zagreb, Croatia
  • Year and Date: 2016-06-01 – 2016-06-01
  • Int'l Joint Research / Invited
• [Presentation] Vertex algebras and symplectic varieties (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: CMSA Special Mathematical Physics Seminar
  • Place of Presentation: Harvard University, USA
  • Year and Date: 2016-05-13 – 2016-05-13
  • Int'l Joint Research / Invited
• [Presentation] Vertex algebras and symplectic varieties (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Algebra Seminar
  • Place of Presentation: University of Zagreb, Croatia
  • Year and Date: 2016-04-22 – 2016-04-22
  • Int'l Joint Research / Invited
• [Presentation] Pentagonal number theorem and representation theory (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Colloquium
  • Place of Presentation: University of Virginia, USA
  • Year and Date: 2016-04-21 – 2016-04-21
  • Int'l Joint Research / Invited
• [Presentation] Deligne exceptional series, Feigin-Frenkel conjecture, and W-algebras (2016)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Representation theory and symplectic singularities in honour of Sasha Premet’s 60th birthday,
  • Place of Presentation: ICMS, Edinburgh, UK
  • Year and Date: 2016-04-04 – 2016-04-04
  • Int'l Joint Research / Invited
• [Remarks] 荒川 知幸
  • URL: http://www.kurims.kyoto-u.ac.jp/~arakawa/