Does a compact hyperbolic 4-manifold have a symplectic structure?

• Project/Area Number: 17K14186
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Nagoya University
• Principal Investigator: Matsuo Shinichiroh 名古屋大学, 多元数理科学研究科, 准教授 (40599487)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 幾何解析 / ゲージ理論 / 微分幾何 / 大域解析 / 指数定理 / 格子ゲージ理論 / アノマリー / 四次元多様体 / 双曲計量 / シンプレクティック構造

Outline of Final Research Achievements

We have investigated the scope of Witten localisation techniques:
First, we studied the relation between the index on manifolds with boundary and domain wall fermions, inspired by lattice gauge theory; and we derived a formula expressing the Atiyah-Patodi-Singer index interms of the eta invariants of domain wall fermion Dirac operators. We also studied mod 2 extensions.
Second, we have studied the orientation of moduli spaces of anti-self-dual metrics.

Academic Significance and Societal Importance of the Research Achievements

指数定理と格子ゲージ理論を架橋し，Atiyah-Patodi-Singerの指数定理の格子ゲージ理論的定式化に道を開いた．さらに，指数の局所化の射程を探った．第一に，境界付き多様体の指数とドメインウォールフェルミオンとの関係を，物理の格子ゲージ理論に触発され，研究したことがある．その結果，APS指数をドメインウォールフェルミオンのエータ不変量で表す公式を得た．これらの公式のmod 2指数や複素フェルミオンへの拡張も研究した．第二に，反自己双対計量のモジュライ空間の向き付け可能性を判定するためのKR指数の計算がある．

Research Products

• [Journal Article] Mod-two APS index and domain-wall fermion (2022)
  • Author(s): Fukaya Hidenori、Furuta Mikio、Matsuki Yoshiyuki、Matsuo Shinichiroh、Onogi Tetsuya、Yamaguchi Satoshi、Yamashita Mayuko
  • Journal Title: Letters in Mathematical Physics Volume: 112 Issue: 2 Pages: 1-16
  • DOI: 10.1007/s11005-022-01509-2
  • Peer Reviewed / Open Access
• [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: Communicatios in Mathematical Physics Volume: 380 Issue: 3 Pages: 1295-1311
  • DOI: 10.1007/s00220-020-03806-0
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Mod-two APS index and domain-wall fermion (2021)
  • Author(s): 松尾 信一郎
  • Organizer: International Conference on Discrete Geometric Analysis for Materials Design
  • Int'l Joint Research / Invited
• [Presentation] The APS index theorem, domain-wall fermions, and global anomaly inflow (2021)
  • Author(s): 松尾 信一郎
  • Organizer: SUSTech-Nagoya workshop on Quantum Science
  • Int'l Joint Research / Invited
• [Presentation] 境界付き多様体の Atiyah-Patodi-Singer の指数とドメインウォールフェルミオン (2020)
  • Author(s): 松尾信一郎
  • Organizer: 東大数理 トポロジー火曜セミナー
  • Invited
• [Presentation] Atiyah-Patodi-Singerの指数定理とドメインウォールフェルミオンの数学 (2020)
  • Author(s): 松尾信一郎
  • Organizer: Summer School 数理物理 2020 指数定理の数理
  • Invited
• [Presentation] The Atiyah-Patodi-Singer index and domain-wall fermion Dirac operators (2020)
  • Author(s): 松尾信一郎
  • Organizer: The Eighth Pacific Rim Conference in Mathematics
  • Int'l Joint Research / Invited
• [Presentation] The Atiyah-Patodi-Singer index theorem and domain walls (2019)
  • Author(s): 松尾信一郎
  • Organizer: NTTI 2019 and BEC 2019
  • Int'l Joint Research / Invited
• [Presentation] Scalar curvature and the twisted Seiberg-Witten equation (2018)
  • Author(s): 松尾 信一郎
  • Organizer: The Joint International Meeting of the Chinese Mathematical Society and the American Mathematical Society
  • 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
  • Int'l Joint Research / Invited
• [Presentation] Scalar curvature and the twisted Seiberg-Witten equations (2017)
  • Author(s): Shinichiroh Matsuo
  • Organizer: PRIMA third congress at Oaxaca, Mexico
  • Int'l Joint Research / Invited
• [Presentation] Brody curves and mean dimension (2017)
  • Author(s): Shinichiroh Matsuo
  • Organizer: Mean Dimension and Sofic Entropy Meet Dynamical Systems, Geometric Analysis and Information Theory
  • Int'l Joint Research / Invited
• [Remarks] 松尾信一郎のホームページ
  • URL: https://www.math.nagoya-u.ac.jp/~shinichiroh/