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Gauge theory on noncommutative Kähler manifolds constructed by deformation quantization

Research Project

Project/Area Number 16K05138
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionTokyo University of Science

Principal Investigator

Sako Akifumi  東京理科大学, 理学部第二部数学科, 教授 (00424200)

Co-Investigator(Kenkyū-buntansha) 長谷部 一気  仙台高等専門学校, 総合工学科, 准教授 (60435469)
Project Period (FY) 2016-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords非可換幾何 / ケーラー多様体 / 場の量子論 / ゲージ理論 / 行列模型 / 変形量子化 / 場の理論 / インスタントン / リッチ平坦 / アインシュタイン多様体 / Seiberg-Witten写像 / 微分幾何 / 位相的場の理論 / トポロジカル絶縁体
Outline of Final Research Achievements

We have shown that deformation quantization on noncommutative Kahler manifolds corresponds to local Fock representations, and have succeeded in constructing a concrete construction method for the Fock representations and the dictionary for translation between the deformation quantizations and Fock representations. In the simplest noncommutative Kahler manifold, the noncommutative Euclidean space (dimensions 2, 4, and 6), the simplest model of field theory, the scalar field theory (Grosse Wulkenhaar model), is constructed as a system of Schinger-Dyson equations, and all n-point functions are exactly solved in the strong noncommutative limit. We also showed that instanton solutions appearing in gauge theories on the four-dimensional noncommutative Euclidean space correspond to Ricci flat measures on Hermitian manifolds.

Academic Significance and Societal Importance of the Research Achievements

非可換ケーラー多様体上の場の量子論の研究が進展した.場の量子論は,現時点で全ての物理理論を記述する方法であり,その解明は人類全体の発展において最重要な問題である.場の量子論は発散の困難のため数学的に厳密に定義できない場合がほとんどであり,さらに場の量子論の目的であるn点関数は摂動計算で近似解が求められることが通常であり,厳密解が得られることはまれである.非可換空間上の場の理論をもとに行列模型を構築する事で,n点関数の方程式の組として場の量子論を構築する例を作り,ある極限では厳密にその方程式を解くことに成功した.他にも非可換ケーラー多様体の構成法やゲージ理論と計量の関係の解明に進展があった.

Report

(6 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (45 results)

All 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (4 results) Journal Article (17 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 17 results,  Open Access: 3 results,  Acknowledgement Compliant: 5 results) Presentation (23 results) (of which Int'l Joint Research: 16 results,  Invited: 8 results) Book (1 results)

  • [Int'l Joint Research] Sogang University(韓国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Sogang University(韓国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Vienna University(Austria)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Wilhelms University(Germany)

    • Related Report
      2017 Research-status Report
  • [Journal Article] A Generalization of the Quantization of Poisson Manifolds2020

    • Author(s)
      Gohara Jumpei、Marin Drinov Academic Publishing House、Hirota Yuji、Sako Akifumi
    • Journal Title

      Geometry Integrability and Quantization

      Volume: 21 Pages: 138-148

    • DOI

      10.7546/giq-21-2020-138-148

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Categorical perspective on quantization of Poisson algebra2020

    • Author(s)
      Gohara Jumpei、Hirota Yuji、Sako Akifumi
    • Journal Title

      Journal of Mathematical Physics

      Volume: 61 Issue: 7 Pages: 073506-073506

    • DOI

      10.1063/1.5145262

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] SO(5) Landau models and nested Nambu matrix geometry2020

    • Author(s)
      Hasebe Kazuki
    • Journal Title

      Nuclear Physics B

      Volume: 956 Pages: 115012-115012

    • DOI

      10.1016/j.nuclphysb.2020.115012

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A unified construction of Skyrme-type non-linear sigma models via the higher dimensional Landau models2020

    • Author(s)
      Hasebe Kazuki
    • Journal Title

      Nuclear Physics B

      Volume: 961 Pages: 115250-115250

    • DOI

      10.1016/j.nuclphysb.2020.115250

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] $Sp(4; \mathbb{R})$ squeezing for Bloch four-hyperboloid via the non-compact Hopf map2020

    • Author(s)
      Hasebe Kazuki
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 53 Issue: 5 Pages: 055303-055303

    • DOI

      10.1088/1751-8121/ab3cda

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Hermitian-Einstein metrics from noncommutative U(1) instantons2019

    • Author(s)
      Hara Kentaro、Sako Akifumi、Yang Hyun Seok
    • Journal Title

      Journal of Mathematical Physics

      Volume: 60 Issue: 9 Pages: 092501-092501

    • DOI

      10.1063/1.5063528

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] SO(4) Landau models and matrix geometry2018

    • Author(s)
      Hasebe Kazuki
    • Journal Title

      Nuclear Physics B

      Volume: 934 Pages: 149-211

    • DOI

      10.1016/j.nuclphysb.2018.06.023

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Quantization of Locally Symmetric Kahler Manifolds2018

    • Author(s)
      Hara Kentaro、Sako Akifumi
    • Journal Title

      Geometry Integrability and Quantization

      Volume: 19 Pages: 122-131

    • DOI

      10.7546/giq-19-2018-122-131

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] The Φ_4^3 and Φ_6^3 matricial QFT models have reflection positive two-point function2018

    • Author(s)
      Grosse Harald、Sako Akifumi、Wulkenhaar Raimar
    • Journal Title

      NUCLEAR PHYSICS B

      Volume: 926 Pages: 20-48

    • DOI

      10.1016/j.nuclphysb.2017.10.022

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Noncommutative Deformations of Locally Symmetric Kahler manifolds2017

    • Author(s)
      Kentaro Hara, Akifumi Sako
    • Journal Title

      Journal of Geometry and Physics

      Volume: 114 Pages: 554-569

    • DOI

      10.1016/j.geomphys.2017.01.009

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] A Recipe To Construct A Gauge Theory On A Noncommutative Kahler Manifold2017

    • Author(s)
      Sako Akifumi
    • Journal Title

      Noncommutative Geometry and Physics

      Volume: 4 Pages: 361-404

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Fock Representations and Deformation Quantization of Kahler Manifolds2017

    • Author(s)
      Akifumi Sako, Hiroshi Umetsu
    • Journal Title

      Advances in Applied Clifford Algebras

      Volume: - Issue: 3 Pages: 2769-2794

    • DOI

      10.1007/s00006-016-0753-z

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Exact solution of matricial Φ_2^3 quantum field theory2017

    • Author(s)
      Grosse Harald、Sako Akifumi、Wulkenhaar Raimar
    • Journal Title

      NUCLEAR PHYSICS B

      Volume: 925 Pages: 319-347

    • DOI

      10.1016/j.nuclphysb.2017.10.010

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Deformation Quantization of Kahler Manifolds and Their Twisted Fock Representation2017

    • Author(s)
      Akifumi Sako, Hiroshi Umetsu
    • Journal Title

      Geometry, Integrability and Quantization XVII, Proceedings of the Eighteenth International Conference on Geometry, Integrability and Quantization XVIII

      Volume: XVII Pages: 225-240

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Twisted Fock Representations of Noncommutative Kahler Manifolds2016

    • Author(s)
      Akifumi Sako, Hiroshi Umetsu
    • Journal Title

      Journal of Mathematical Physics

      Volume: 57 Issue: 9

    • DOI

      10.1063/1.4961930

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Relastivistic Landau models and generation of fuzzy spheres2016

    • Author(s)
      Kazuki Hasebe
    • Journal Title

      International Journal of Modern Physics A

      Volume: 31 Issue: 20n21 Pages: 1650117-1650117

    • DOI

      10.1142/s0217751x16501177

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Formulation of the relativistic quantum Hall effect and parity anomaly2016

    • Author(s)
      Kouki Yonaga, Kazuki Hasebe, Naokazu Shibata
    • Journal Title

      Physical Review B

      Volume: 93

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Phi^3 matrix model and its exact solution2020

    • Author(s)
      Akifumi Sako
    • Organizer
      4th Bangkok workshop on Discrete Geometry and Statics
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A Generalization of the Quantization of Poisson Manifolds2019

    • Author(s)
      Jumpei Gohara, Yuji Hirota, Akifumi Sako
    • Organizer
      XXIth International Conference on Geometry, Integrability and Quantization
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] A Generalization of the Quantization of Poisson Manifolds2019

    • Author(s)
      Jumpei Gohara, Yuji Hirota, Akifumi Sako
    • Organizer
      The XXVIth International Conference on Integrable Systems and Quantum symmetries (ISQS-26)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] 幾何学としての量子化の一般化2019

    • Author(s)
      郷原惇平、佐古彰史
    • Organizer
      離散的手法による場と時空のダイナミクス2019
    • Related Report
      2019 Research-status Report
  • [Presentation] 量子化の一般化のような話題2019

    • Author(s)
      郷原惇平、佐古彰史、廣田祐士
    • Organizer
      科学基礎論夏のセミナー2019
    • Related Report
      2019 Research-status Report
  • [Presentation] Twisted Fock representation of Kähler manifolds and Hermitian-Einstein metrics from noncommutative U(1) instantons2018

    • Author(s)
      Akifumi Sako, Hiroshi Umetsu, Kentaro Hara, Hyun Seok Yang
    • Organizer
      HOMOTOPY ALGEBRAS, DEFORMATION THEORY AND QUANTIZATION
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Einstein metrics from NC U(1) instantons2018

    • Author(s)
      Kentaro Hara, Akifumi Sako, Hyun Seok Yang
    • Organizer
      GROUP32 (The 32nd International Colloquium on Group Theoretical Methods in Physics)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Non-commutative Geometry and differential topology in higher dimensional Landau models2018

    • Author(s)
      Hasebe Kazuki
    • Organizer
      International conference on holography, string theory and discrete approaches in Hanoi
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Noncommutative Kahler manifolds -from Karabegov's deformation quantization -2018

    • Author(s)
      Akifumi Sako
    • Organizer
      Koriyama Geometry and Physics Days 2018 "Noncommutative geometry and related topics"
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Quantum scalar field theories on noncommutative R2, R4 and R6 as solvable models2018

    • Author(s)
      Akifumi Sako
    • Organizer
      Koriyama Geometry and Physics Days 2018 "Noncommutative geometry and related topics"
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Φ3模型の量子論の行列模型を通した定式化と厳密解2017

    • Author(s)
      佐古彰史
    • Organizer
      日本数学会年会
    • Place of Presentation
      大阪大学
    • Year and Date
      2017-03-26
    • Related Report
      2016 Research-status Report
  • [Presentation] モヤル空間上のΦ3模型の強非可換極限におけるN 点関数の厳密解2017

    • Author(s)
      佐古彰史
    • Organizer
      日本物理学会 年会
    • Place of Presentation
      大阪大学
    • Year and Date
      2017-03-20
    • Related Report
      2016 Research-status Report
  • [Presentation] Φ3 模型の量子論としての定式化への試みとその厳密解2017

    • Author(s)
      佐古彰史
    • Organizer
      多弦数理物理学セミナー
    • Place of Presentation
      名古屋大学
    • Year and Date
      2017-01-27
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Φ^3 model - from a matrix model to a field theory -2017

    • Author(s)
      Akifumi Sako
    • Organizer
      The XXVth International Conference on Integrable Systems and Quantum symmetries (ISQS-25)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Φ^3行列模型:可解な新しい場の量子論のおもちゃ2017

    • Author(s)
      佐古彰史
    • Organizer
      東京理科大学 4学科合同セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] ファイ3乗行列模型の解と場の理論2017

    • Author(s)
      佐古彰史
    • Organizer
      仙台高専 特別講義
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Exact Solution of Noncommutative Phi^3 Model2017

    • Author(s)
      Akifumi Sako
    • Organizer
      Discrete Approaches to the Dynamics of Fields and Space-Time
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 公理論的場の理論の可能性のある2,4,6次元の可解 Phi^3 模型2017

    • Author(s)
      佐古彰史
    • Organizer
      科学基礎論夏のセミナー2017
    • Related Report
      2017 Research-status Report
  • [Presentation] Fock representations and deformation quantization of Kahler manifolds2016

    • Author(s)
      Akifumi Sako, Hiroshi Umetsu
    • Organizer
      Alterman Conference on Geometric Algebra and Summer School on Kahler Calculus
    • Place of Presentation
      Transilvania University of Brasov (Romania)
    • Year and Date
      2016-08-06
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Noncommutative Kahler manifolds and field theory2016

    • Author(s)
      Akifumi Sako
    • Organizer
      Seminar of Division of Theoretical Physics
    • Place of Presentation
      Ruder Boskovic Institute (Zagreb, Croatia)
    • Year and Date
      2016-07-12
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Noncommutative Kahler Manifolds and their Fock representations2016

    • Author(s)
      Akifumi Sako
    • Organizer
      VI Congress of Mathematicains of Macedonia 2016
    • Place of Presentation
      Hotel Metropol(Macedonia)
    • Year and Date
      2016-06-17
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Deformation Quantization of Kahler Manifolds and Their Twisted Fock Representation2016

    • Author(s)
      Akifumi Sako, Hiroshi Umetsu
    • Organizer
      XVIIIth International Conference Geometry, Integrability and Quantization
    • Place of Presentation
      Hotel Koral (Varna, Bulgaria)
    • Year and Date
      2016-06-03
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Noncommutative Kahler manifolds and field theory2016

    • Author(s)
      Akifumi Sako
    • Organizer
      Seminar fur Mathematische Physik
    • Place of Presentation
      Vienna University
    • Year and Date
      2016-05-24
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] 非可換微分幾何学の基礎2020

    • Author(s)
      前田 吉昭、佐古 彰史
    • Total Pages
      296
    • Publisher
      共立出版
    • ISBN
      9784320112070
    • Related Report
      2020 Annual Research Report

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Published: 2016-04-21   Modified: 2022-01-27  

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