指数定理の展開とトポロジカル表面状態

• Project Area: Discrete Geometric Analysis for Materials Design
• Project/Area Number: 17H06461
• Research Category: Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area)
• Allocation Type: Single-year Grants
• Review Section: Science and Engineering
• Research Institution: The University of Tokyo
• Principal Investigator: 古田 幹雄 東京大学, 大学院数理科学研究科, 教授 (50181459)
• Co-Investigator(Kenkyū-buntansha): 五味 清紀 東京工業大学, 理学院, 教授 (00543109) ; 加藤 毅 京都大学, 理学研究科, 教授 (20273427)
• Project Period (FY): 2017-06-30 – 2022-03-31
• Project Status: Granted (Fiscal Year 2021)
• Budget Amount: ¥52,130,000 (Direct Cost: ¥40,100,000、Indirect Cost: ¥12,030,000); Fiscal Year 2021: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000); Fiscal Year 2020: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000); Fiscal Year 2019: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000); Fiscal Year 2018: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000); Fiscal Year 2017: ¥5,330,000 (Direct Cost: ¥4,100,000、Indirect Cost: ¥1,230,000)
• Keywords: トポロジー / Bott周期性 / K群 / スペクトル切断 / 格子指数定理 / Wilson-Dirac作用素 / 指数定理 / ドメインウォール / ゲージ理論

Outline of Annual Research Achievements

Atiyah-Singerの指数定理は閉多様体上の線形偏微分方程式の解空間の次元から決まる「指数」を、ある条件の下でトポロジカルに与える公式である。 本研究のひとつの課題は、ある設定の下で、指数定理を（１）非線形偏微分作用素に対して、また（２）閉ではない多様体、に対して拡張し、ある種の非線形偏微 分作用素に対して解の散乱に相当する現象をトポロジカルに把握することである。ここで（１）としては３，４多様体上のゲージ理論の方程式、（２）としては 端のモデルが閉多様体の被覆空間である開多様体が対象である。
・昨年度の主要な成果は格子ゲージ理論におけるWilson-Dirac作用素の指数と連続理論におけるDirac作用素の指数を結び付けたことであったが、今年度は、前者をK理論的に定式化する記述方法を整備した。
・今年度の主要な成果として、K理論において基本的なBott周期性の、スペクトル切断による新しい別証明を与えた。数年前に、物性物理のバルクエッジ対応の考察にヒントを得てBott周期性のいわば"量子力学的証明"を得ていた。今回は、その証明から無限次元的考察を排除し、完全に有限次元の線形代数の範疇で証明を与えた。S^1上の微分作用素に対して、スペクトル切断の明示的な構成を行い、有限次元近似構成の基本的手段として用いる。この構成は、複素K群の周期性のみならず、実のKO群、あるいはKR理論の非自明な懸垂についての周期性に直接拡張可能である。
・上のスペクトル切断の構成の動機は上記（１）での有限次元近似への応用である。しかしこれについては今年度は明確な進展はなかった。
・物理学者の深谷英則・大野木哲也・山口哲の3氏および数学者の松尾信一郎・山下真由子の2氏との共同研究を進め。Atiyah-Patodi-Singer境界条件におけるmod 2 指数のドメインウォールを用いた具体的な記述を与えた。

Current Status of Research Progress

3: Progress in research has been slightly delayed.

Reason

●今年度は予定していた国際研究集会の計画をコロナの影響で取りやめた。それに限らず、3人以上のメンバーが一堂に会して広い黒板を前にして対面で議論を行う機会が持てなかった。代表者はオンラインでの講義の準備等に予想以上の労力が必要であった。代表者は本研究を進めるために「K理論的な定式化の記述方法の整備」のプロジェクトは、大学院向けの講義を用いた発表と軌を一にして行ったが、十分には行えなかった。
●昨年度に引き続く物理学者と数学者との共同研究は、準定期的なオンラインミーティングによって行った。対面とは質の異なる議論となり、長時間連続して行うことは難しかった。
●本研究の非線形作用素とかかわる側面については、新しい可能性のアイディアは得たが（スペクトル切断を"effective"に構成することによる可能性）結果的には、その実行を試みるまでには至らなかった。

Strategy for Future Research Activity

●2020年度に開催できなかった国際研究集会の代替の集会を2021年度に開催したい。オンラインが適切か否かを検討する。
●これまで若手の協力研究員を適宜追加し、若手人材の育成を行ってきた。概ねこの方針は有効に働いてきたと思われる。この方針を地道に堅持する。

Research Products

• [Journal Article] Homological bulk-edge correspondence for Weyl semimetal (2021)
  • Author(s): Kiyonori Gomi
  • Journal Title: Progress of Theoretical and Experimental Physics Volume: 0 Pages: 0-0
  • DOI: 10.1093/ptep/ptab035
  • Peer Reviewed / Open Access
• [Journal Article] The simple type conjecture for mod 2 Seiberg-Witten invariants (2021)
  • Author(s): Kato Tsuyoshi、 Nakamura Nobuhiro、 Yasui Koichi
  • Journal Title: Journal of the European Mathematical Society Volume: -
  • Peer Reviewed
• [Journal Article] Freed-Moore K-theory (2021)
  • Author(s): Kiyonori Gomi
  • Journal Title: Communications in Analysis and Geometry Volume: -
  • Peer Reviewed
• [Journal Article] A note on the Nielsen realization problem for K3 surfaces (2021)
  • Author(s): David Baraglia, Hokuto Konno
  • Journal Title: Proceedings of the American Mathematical Society Volume: -
  • DOI: 10.1090/proc/15544
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Kitaev’s Stabilizer Code and Chain Complex Theory of Bicommutative Hopf Algebras (2021)
  • Author(s): Kim Minkyu
  • Journal Title: Communications in Mathematical Physics Volume: -
  • Peer Reviewed
• [Journal Article] The Atiyah-Patodi-Singer Index and Domain-Wall Fermion Dirac Operators (2020)
  • Author(s): H. Fukaya, M. Furuta, S. Matsuo, T. Onogi, S. Yamaguchi and M. Yamashita
  • Journal Title: Communications in Mathematical Physics Volume: 380 Pages: 1295-1311
  • DOI: 10.1007/s00220-020-03806-0
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Smoothness filtration of the magnitude complex (2020)
  • Author(s): Gomi Kiyonori
  • Journal Title: Forum Mathematicum Volume: 32 Pages: 625-639
  • DOI: 10.1515/forum-2019-0091
  • Peer Reviewed
• [Journal Article] Eigenvalue crossings in Floquet topological systems (2020)
  • Author(s): Gomi Kiyonori、Tauber Clement
  • Journal Title: Letters in Mathematical Physics Volume: 110 Pages: 465-500
  • DOI: 10.1007/s11005-019-01225-4
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The Geometry of (non-Abelian) Landau levels (2020)
  • Author(s): Giuseppe De Nittis, Kiyonori Gomi, Massimo Moscolari
  • Journal Title: Journal of Geometry and Physics Volume: 152 Pages: 103649-103649
  • DOI: 10.1016/j.geomphys.2020.103649
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Positive scalar curvature and 10/8-type inequalities on 4-manifolds with periodic ends (2020)
  • Author(s): Hokuto Konno, Masaki Taniguchi
  • Journal Title: Inventiones mathematicae Volume: 222 Pages: 833-880
  • DOI: 10.1007/s00222-020-00979-2
  • Peer Reviewed
• [Journal Article] On the K-theoretic classification of dynamically stable systems (2019)
  • Author(s): Giuseppe De Nittis, Kiyonori Gomi
  • Journal Title: Reviews in Mathematical Physics Volume: 31 Pages: 1950003-1950003
  • DOI: 10.1142/s0129055x1950003x
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices (2019)
  • Author(s): Kiyonori Gomi,Guo Chuan Thiang
  • Journal Title: Letters in Mathematical Physics Volume: 109 Pages: 857-904
  • DOI: 10.1007/s11005-018-1129-1
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Crystallographic T-duality (2019)
  • Author(s): Kiyonori Gomi,Guo Chuan Thiang
  • Journal Title: Journal of Geometry and Physics Volume: 139 Pages: 50-77
  • DOI: 10.1016/j.geomphys.2019.01.002
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Second-order topological phases protected by chiral symmetry (2019)
  • Author(s): Okugawa Ryo、Hayashi Shin、Nakanishi Takeshi
  • Journal Title: Physical Review B Volume: 100 Pages: 205302-205302
  • DOI: 10.1103/physrevb.100.235302
  • Peer Reviewed
• [Journal Article] Toeplitz operators on concave corners and topologically protected corner states (2019)
  • Author(s): Hayashi Shin
  • Journal Title: Letters in Mathematical Physics Volume: 109 Pages: 2223-2254
  • DOI: 10.1007/s11005-019-01184-w
  • Peer Reviewed
• [Journal Article] Chiral vector bundles (2018)
  • Author(s): Giuseppe De Nittis, Kiyonori Gomi
  • Journal Title: Mathematische Zeitschrift Volume: 290 Pages: 775-830
  • DOI: 10.1007/s00209-018-2041-1
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Topological Invariants and Corner States for Hamiltonians on a Three-Dimensional Lattice (2018)
  • Author(s): Hayashi Shin
  • Journal Title: Communications in Mathematical Physics Volume: 364 Pages: 343-356
  • DOI: 10.1007/s00220-018-3229-2
  • Peer Reviewed
• [Journal Article] Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries (2018)
  • Author(s): Ken Shiozaki, Hassan Shapourian, Kiyonori Gomi, Shinsei Ryu
  • Journal Title: Physical Review B Volume: 98
  • DOI: 10.1103/physrevb.98.035151
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The FKMM-invariant in low dimension (2018)
  • Author(s): De Nittis Giuseppe、Gomi Kiyonori
  • Journal Title: Letters in Mathematical Physics Volume: 108 Pages: 1225-1277
  • DOI: 10.1007/s11005-017-1029-9
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The cohomological nature of the Fu-Kane-Mele invariant (2018)
  • Author(s): Guseppe De Nittis, Kiyonori Gomi
  • Journal Title: Journal of Geometry and Physics Volume: 124 Pages: 124-164
  • DOI: 10.1016/j.geomphys.2017.10.007
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Fu-Kane-Mele monopoles in semimetals (2017)
  • Author(s): Guo Chuan Thiang, Koji Sato and Kiyonori Gomi
  • Journal Title: Nuclear Physics B Volume: 923 Pages: 107-125
  • DOI: 10.1016/j.nuclphysb.2017.07.018
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Twists on the torus equivariant under the 2-dimensional crystallographic point groups (2017)
  • Author(s): Kiyonori Gomi
  • Journal Title: Symmetry, Integrability and Geometry: Methods and Applications Volume: 13 Pages: 014-051
  • DOI: 10.3842/sigma.2017.014
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization III (2017)
  • Author(s): Yoshiko Ogata
  • Journal Title: Communications in Mathematical Physics Volume: 352 Pages: 1205-1263
  • DOI: 10.1007/s00220-016-2810-9
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Bulk-edge correspondence and the cobordism invariance of the index (2017)
  • Author(s): Hayashi Shin
  • Journal Title: Reviews in Mathematical Physics Volume: 29 Pages: 1750033-1750033
  • DOI: 10.1142/s0129055x17500337
  • Peer Reviewed / Int'l Joint Research
• [Presentation] 同変K理論によるバンド絶縁体の分類 (2021)
  • Author(s): 五味 清紀
  • Organizer: オンラインCREST チュートリアル・ワークショップ 物質のトポロジカル相の理論的探究
  • Invited
• [Presentation] Topological invariants and corner states for Hamiltonians on a lattice (2021)
  • Author(s): 林 晋
  • Organizer: 第27回大阪市立大学国際学術シンポジウム 可視化の数理と対称性およびモジュライの深化
  • Invited
• [Presentation] トーラス上のDirac作用素の正方格子による有限次元近似 (2020)
  • Author(s): 古田幹雄
  • Organizer: Summer School 数理物理 2020
  • Invited
• [Presentation] L^2 harmonic theory and Seiberg-Witten・Bauer-Furuta theory on non-compact complete Riemannian 4-manifolds (2020)
  • Author(s): Kato Tsuyoshi
  • Organizer: The Eighth Pacific Rim Conference in Mathematics, Session on Differential Geometry
  • Int'l Joint Research / Invited
• [Presentation] L^2 harmonic theory and Seiberg-Witten・Bauer-Furuta theory on non-compact complete Riemannian 4-manifolds (2020)
  • Author(s): Kato Tsuyoshi
  • Organizer: SouthamptonKyoto Workshop 2
  • Int'l Joint Research / Invited
• [Presentation] A mathematical approach to higher-order topological insulators (2020)
  • Author(s): 林 晋
  • Organizer: トポロジカル表面状態、ソリトンとブレーン、指数定理
  • Invited
• [Presentation] Toeplitz作用素の指数理論と(高次)トポロジカル絶縁体のトポロジーについて (2020)
  • Author(s): 林 晋
  • Organizer: 神戸幾何学セミナー, 神戸大学
  • Invited
• [Presentation] Index of the Wilson-Dirac operator revisited: a discrete version of Dirac operator on a finite lattice (2020)
  • Author(s): 古田 幹雄
  • Organizer: 理研 iTHEMS Math Seminar
  • Invited
• [Presentation] The Atiyah-Patodi-Singer index theorem and domain walls (2019)
  • Author(s): Furuta Mikio
  • Organizer: MATERIALS 56 RESEARCH MEETING 2019 Materials Innovation for Sustainable Development Goals, YOKOHAMA SYMPOSIA
  • Int'l Joint Research / Invited
• [Presentation] The Atiyah-Patodi-Singer index theorem and domain walls (2019)
  • Author(s): Furuta Mikio
  • Organizer: International Molecule-type Workshop Frontiers in Lattice QCD and related topics, Yukawa Institute for Theoretical Physics, Kyoto University
  • Int'l Joint Research / Invited
• [Presentation] Seiberg-Witten 理論周辺の応用と発展 (2019)
  • Author(s): 加藤毅
  • Organizer: 第６６回幾何学シンポジウム
  • Invited
• [Presentation] A rigidity theorem of the Z_2-valued Seiberg-Witten invariants for spin families (2019)
  • Author(s): Kato Tsuyoshi
  • Organizer: Index Theory, Duality and Related FieldsIn honour of Varghese Mathai’s 60th birthday
  • Int'l Joint Research / Invited
• [Presentation] Twisted Donaldson invariant (2019)
  • Author(s): Kato Tsuyoshi
  • Organizer: K-theory and Noncommutative Geometry
  • Int'l Joint Research / Invited
• [Presentation] Topological Phases of Interacting Quantum Systems (2019)
  • Author(s): Gomi Kiyonori
  • Organizer: Casa Matematica Oaxaca, Mexico
  • Int'l Joint Research / Invited
• [Presentation] Topological phase and generalized (co)homology (2019)
  • Author(s): Gomi Kiyonori
  • Organizer: Rigorous Statistical Mechanics and Related Topics, Kyoto University
  • Int'l Joint Research / Invited
• [Presentation] Topological invariants and corner states for Hamiltonians on a lattice (2019)
  • Author(s): 林 晋
  • Organizer: 第６６回トポロジーシンポジウム
  • Invited
• [Presentation] ある種の角に対するテープリッツ作用素の指数理論とその応用 (2019)
  • Author(s): 林 晋
  • Organizer: 第６６回幾何学シンポジウム
  • Invited
• [Presentation] Topological phases of interacting quantum systems (2019)
  • Author(s): Hayashi Shin
  • Organizer: Casa Matematica Oaxaca, Mexico
  • Int'l Joint Research / Invited
• [Presentation] The Atiyah-Patodi-Singer index theorem and domain walls (2018)
  • Author(s): 古田幹雄
  • Organizer: BEC2018X Variety and universality of bulk-edge correspondence in topological phases: From solid state physics to transdisciplinary concepts, Tsukuba University
  • Int'l Joint Research / Invited
• [Presentation] Topological invariants andcorner states for Hamiltonians on a three dimensional lattice (2018)
  • Author(s): 林晋
  • Organizer: Topology and computer 2018, Nara Women’s University
  • Int'l Joint Research / Invited
• [Presentation] Twisted Donaldson invariant (2018)
  • Author(s): 加藤毅
  • Organizer: Gauge Theory and Applications, Universität Regensburg
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic volume growth of the instanton moduli spaces over the four-sphere (2018)
  • Author(s): 松尾信一郎
  • Organizer: Gauge theory and applications at Universität Regensburg
  • Int'l Joint Research / Invited
• [Presentation] Band topology and submanifolds of matrices (2018)
  • Author(s): 五味清紀
  • Organizer: Recent progress in mathematics of topological insulators, ETH Zürich, Switzerland
  • Int'l Joint Research / Invited
• [Presentation] Twisted Donaldson invariant (2018)
  • Author(s): 加藤毅
  • Organizer: Gauge Theory in Fukuoka
  • Int'l Joint Research / Invited
• [Presentation] Relative higher index and almost flat bundles (2018)
  • Author(s): 窪田陽介
  • Organizer: Gauge Theory in Fukuoka
  • Int'l Joint Research / Invited
• [Presentation] A (3+1)-dimensional TQFT with TRS-like symmetry (2018)
  • Author(s): 古田幹雄
  • Organizer: Variety and universality of bulk-edge correspondence in topological phases, University of Tsukuba
  • Int'l Joint Research / Invited
• [Presentation] Topological crystalline insulators and K-theory (2018)
  • Author(s): 五味清紀
  • Organizer: Topology & Condensed Matter , National Tsing Hua University, Taiwan, March 22-23
  • Int'l Joint Research / Invited
• [Presentation] Pin(2)-Seiberg-Witten Floer functor on homology 3-spheres (2017)
  • Author(s): 古田幹雄
  • Organizer: Mathematical Aspects of Topological Phases of Matter and Quantum Computing, AIMR
  • Int'l Joint Research / Invited
• [Presentation] A class of asymmetric gapped Hamiltonians on quantum spin (2017)
  • Author(s): 緒方芳子
  • Organizer: "Operator algebras and Quantum Information Theory", IHP Paris
  • Int'l Joint Research / Invited
• [Remarks] A01「トポロジカル物質」
  • URL: https://www.math-materials.jp/outline/#a01
• [Remarks] 研究成果報告書
  • URL: https://www.ms.u-tokyo.ac.jp/activity/annualreport.html
• [Remarks] 研究成果報告書 令和元年度 東京大学大学院数理科学研究科
  • URL: https://www.ms.u-tokyo.ac.jp/activity/documents/AnnualReport2019.pdf
• [Remarks] 東大数理： 研究成果報告書平成30年度
  • URL: https://www.ms.u-tokyo.ac.jp/activity/documents/AnnualReport2018.pdf
• [Remarks] 京都大学 教育研究活動データベース
  • URL: https://kyouindb.iimc.kyoto-u.ac.jp/j/cA2iC
• [Remarks] Kiyonori Gomi's Website
  • URL: http://www.math.titech.ac.jp/~kgomi/
• [Funded Workshop] PROGRESS IN THE MATHEMATICS OF TOPOLOGICAL STATES OF MATTER, AIMR (2019)
• [Funded Workshop] Topology and Computer 2018, Oct. 12 (Fri) - 14 (Sun) around midday Venue: Nara Women’s University, Collaboration Center Z306 (2018)
• [Funded Workshop] Symposium on Transformation Groups, Kumamoto City International Center,December 6(Thu) - December 8(Sat), 2018 (2018)
• [Funded Workshop] East Asian Conference on Gauge theory and Related Topics, Kyoto university, September 11- September 15, 2018 (2018)
• [Funded Workshop] Gauge Theory in Fukuoka (2018)