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Yuji Terashima

Section

Graduate School of Science

Job title

Professor

Degree

Doctor of Mathematical Science

researchmap

https://researchmap.jp/read0163459

Research Interests 1

Topology, Mathematical Physics

Research Areas 1

Natural sciences / Geometry /

Papers 36

Homological blocks with simple Lie algebras and Witten-Reshetikhin-Turaev invariants

Yuya Murakami, Yuji Terashima

to appear in Communications in Mathematical Physics　2026/04
Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

Rei Inoue, Atsuo Kuniba, Xiaoyue Sun, Yuji Terashima, Junya Yagi

Symmetry, Integrability and Geometry: Methods and Applications　2024/12/21
Publisher: SIGMA (Symmetry, Integrability and Geometry: Methods and Application)
DOI： 10.3842/sigma.2024.113 　

eISSN： 1815-0659

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We construct a new solution to the tetrahedron equation by further pursuing the quantum cluster algebra approach in our previous works. The key ingredients include a symmetric butterfly quiver attached to the wiring diagrams for the longest element of type $A$ Weyl groups and the implementation of quantum $Y$-variables through the $q$-Weyl algebra. The solution consists of four products of quantum dilogarithms. By exploring both the coordinate and momentum representations, along with their modular double counterparts, our solution encompasses various known three-dimensional (3D) $R$-matrices. These include those obtained by Kapranov-Voevodsky (1994) utilizing the quantized coordinate ring, Bazhanov-Mangazeev-Sergeev (2010) from a quantum geometry perspective, Kuniba-Matsuike-Yoneyama (2023) linked with the quantized six-vertex model, and Inoue-Kuniba-Terashima (2023) associated with the Fock-Goncharov quiver. The 3D $R$-matrix presented in this paper offers a unified perspective on these existing solutions, coalescing them within the framework of quantum cluster algebra.
On the Burde–de Rham Theorem for Finitely Presented Pro-p Groups

Yasushi Mizusawa, Ryoto Tange, Yuji Terashima

International Mathematics Research Notices　2025　(1)　2024/12/09
Publisher: Oxford University Press (OUP)
DOI： 10.1093/imrn/rnae267 　

eISSN： 1687-0247

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Abstract We consider the Burde–de Rham theorem for finitely presented pro-$p$ groups under the assumption that the total degrees of all relators are $0$. We also give some concrete examples including higher-dimensional cases under Iwasawa theoretic conditions, and consider some cohomological interpretations.
The Heisenberg double of involutory Hopf algebras and invariants of closed 3–manifolds

Serban Matei Mihalache, Sakie Suzuki, Yuji Terashima

Algebraic & Geometric Topology　24　(7)　3669-3691　2024/12/09
Publisher: Mathematical Sciences Publishers
DOI： 10.2140/agt.2024.24.3669 　

ISSN： 1472-2747

eISSN： 1472-2739
Discrete higher Berry phases and matrix product states

Shuhei Ohyama, Yuji Terashima, Ken Shiozaki

Physical Review B　110　(3)　2024/07
Publisher: American Physical Society (APS)
DOI： 10.1103/physrevb.110.035114 　

ISSN： 2469-9950

eISSN： 2469-9969
Quantum Cluster Algebras and 3D Integrability: Tetrahedron and 3D Reflection Equations

Rei Inoue, Atsuo Kuniba, Yuji Terashima

International Mathematics Research Notices　2024　(16)　11549-11581　2024/06/17
Publisher: Oxford University Press (OUP)
DOI： 10.1093/imrn/rnae128 　

ISSN： 1073-7928

eISSN： 1687-0247

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Abstract We construct a new solution to the tetrahedron equation and the three-dimensional (3D) reflection equation by extending the quantum cluster algebra approach by Sun and Yagi concerning the former. We consider the Fock–Goncharov quivers associated with the longest elements of the Weyl groups of type $A$ and $C$, and investigate the cluster transformations corresponding to changing a reduced expression into a “most distant” one. By devising a new realization of the quantum $y$-variables in terms of $q$-Weyl algebra, the solutions are extracted as the operators whose adjoint actions yield the cluster transformations of the quantum $y$-variables. Explicit formulas of their matrix elements are also derived for some typical representations.
Modular Transformations of Homological Blocks for Seifert Fibered Homology 3-Spheres

Toshiki Matsusaka, Yuji Terashima

Communications in Mathematical Physics　405　(2)　2024/02/19
Publisher: Springer Science and Business Media LLC
DOI： 10.1007/s00220-024-04939-2 　

ISSN： 0010-3616

eISSN： 1432-0916
Tetrahedron equation and quantum cluster algebras

Rei Inoue, Atsuo Kuniba, Yuji Terashima

Journal of Physics A: Mathematical and Theoretical　57　(8)　085202-085202　2024/02/08
Publisher: IOP Publishing
DOI： 10.1088/1751-8121/ad2224 　

ISSN： 1751-8113

eISSN： 1751-8121

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Abstract We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y-variables in terms q-Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev’s R matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when q is specialized to a root of unity.
On Adjoint Homological Selmer Modules for SL(2)-Representations of Knot Groups

Takahiro Kitayama, Masanori Morishita, Ryoto Tange, Yuji Terashima

International Mathematics Research Notices　2023　(23)　19801-19826　2022/09/22
Publisher: Oxford University Press (OUP)
DOI： 10.1093/imrn/rnac255 　

ISSN： 1073-7928

eISSN： 1687-0247

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Abstract We introduce the adjoint homological Selmer module for an $\textrm {SL}_2$-representation of a knot group, which may be seen as a knot theoretic analogue of the dual adjoint Selmer module for a Galois representation. We then show finitely generated torsion-ness of our adjoint Selmer module, which are widely known as conjectures in number theory, and give some concrete examples.
Arithmetic Orr Invariants of Absolute Galois Groups Peer-reviewed

Hisatoshi Kodani, Yuji Terashima

International Mathematics Research Notices　138　2022/06/13
Publisher: Oxford University Press (OUP)
DOI： 10.1093/imrn/rnac138 　

ISSN： 1073-7928

eISSN： 1687-0247

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Abstract Based on the analogies between mapping class groups and absolute Galois groups, we introduce an arithmetic pro-$\ell $ analogue of Orr invariants for a Galois element associated with Galois action on étale fundamental groups of punctured projective lines. At the same time, we also introduce pro-$\ell $ Orr space as an arithmetic analogue of Orr space whose third homotopy group is a target group of Orr invariant. We then determine its rank as $\mathbb {Z}_{\ell }$-module following Igusa–Orr’s computation. Moreover, we investigate its relation with Ellenberg’s obstruction to $\pi _1$-sections associated with lower central series filtration in the context of Grothendieck’s section conjecture.
The colored Jones polynomials as vortex partition functions Peer-reviewed

Masahide Manabe, Seiji Terashima, Yuji Terashima

Journal of High Energy Physics　2021　(12)　2021/12
Publisher: Springer Science and Business Media LLC
DOI： 10.1007/jhep12(2021)197 　

eISSN： 1029-8479

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<title>A<sc>bstract</sc> </title>We construct 3D <inline-formula><alternatives><tex-math>$$ \mathcal{N} $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math></alternatives></inline-formula> = 2 abelian gauge theories on <inline-formula><alternatives><tex-math>$$ \mathbbm{S} $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math></alternatives></inline-formula>² × <inline-formula><alternatives><tex-math>$$ \mathbbm{S} $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math></alternatives></inline-formula>¹ labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones polynomials of knots in <inline-formula><alternatives><tex-math>$$ \mathbbm{S} $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math></alternatives></inline-formula>³. The colored Jones polynomials are obtained as the Wilson loop expectation values along knots in SU(2) Chern-Simons gauge theories on <inline-formula><alternatives><tex-math>$$ \mathbbm{S} $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math></alternatives></inline-formula>³, and then our construction provides an explicit correspondence between 3D <inline-formula><alternatives><tex-math>$$ \mathcal{N} $$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math></alternatives></inline-formula> = 2 abelian gauge theories and 3D SU(2) Chern-Simons gauge theories. We verify, in particular, the applicability of our constructions to a class of tangle diagrams of 2-bridge knots with certain specific twists.
Witten–Reshetikhin–Turaev Function for a Knot in Seifert Manifolds Peer-reviewed

Hiroyuki Fuji, Kohei Iwaki, Hitoshi Murakami, Yuji Terashima

Communications in Mathematical Physics　2021/03
Publisher: Springer Science and Business Media LLC
DOI： 10.1007/s00220-021-03953-y 　

ISSN： 0010-3616

eISSN： 1432-0916
Cluster variables, ancestral triangles and Alexander polynomials Peer-reviewed

Wataru Nagai, Yuji Terashima

Advances in Mathematics　363　106965-106965　2020/03
Publisher: Elsevier BV
DOI： 10.1016/j.aim.2019.106965 　

ISSN： 0001-8708
Hyperbolic 3-manifolds and cluster algebras Peer-reviewed

Kentaro Nagao, Yuji Terashima, Masahito Yamazaki

Nagoya Mathematical Journal　235　1-25　2019/09
Publisher: Cambridge University Press (CUP)
DOI： 10.1017/nmj.2017.39 　

ISSN： 0027-7630

eISSN： 2152-6842

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We advocate the use of cluster algebras and their <inline-formula> <alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0027763017000393_inline1" xlink:type="simple" /><tex-math>$y$</tex-math></alternatives> </inline-formula>-variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster <inline-formula> <alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0027763017000393_inline2" xlink:type="simple" /><tex-math>$y$</tex-math></alternatives> </inline-formula>-variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedra. We also comment on the completeness of hyperbolic structures.
Quiver Mutation Sequences and q-Binomial Identities Peer-reviewed

Akishi Kato, Yuma Mizuno, Yuji Terashima

International Mathematics Research Notices　2018　(23)　7335-7358　2018/12
Publisher: Oxford University Press (OUP)
DOI： 10.1093/imrn/rnx108 　

ISSN： 1073-7928

eISSN： 1687-0247
On certain L-functions for deformations of knot group representations Peer-reviewed

Takahiro Kitayama, Masanori Morishita, Ryoto Tange, Yuji Terashima

Transactions of the American Mathematical Society　370　(5)　3171-3195　2017/11
Publisher: American Mathematical Society (AMS)
DOI： 10.1090/tran/7037 　

ISSN： 0002-9947

eISSN： 1088-6850
p-Johnson homomorphisms and pro-p groups Peer-reviewed

Masanori Morishita, Yuji Terashima

Journal of Algebra　479　102-136　2017/06
Publisher: Elsevier BV
DOI： 10.1016/j.jalgebra.2017.01.028 　

ISSN： 0021-8693
On the universal deformations for ${\rm SL}_2$-representations of knot groups Peer-reviewed

Masanori Morishita, Yu Takakura, Yuji Terashima, Jun Ueki

Tohoku Mathematical Journal　69　(1)　67-84　2017/04
Publisher: Mathematical Institute, Tohoku University
DOI： 10.2748/tmj/1493172129 　

ISSN： 0040-8735
Arithmetic Topology in Ihara Theory Peer-reviewed

Hisatoshi Kodani, Masanori Morishita, Yuji Terashima

Publications of the Research Institute for Mathematical Sciences　53　(4)　629-688　2017
Publisher: European Mathematical Society Publishing House
DOI： 10.4171/prims/53-4-6 　

ISSN： 0034-5318
Characteristic classes of fiber bundles Peer-reviewed

Takahiro Matsuyuki, Yuji Terashima

Algebraic and Geometric Topology　16　(5)　3029-3050　2016

DOI： 10.2140/agt.2016.16.3029 　

ISSN： 1472-2739
Quantum Dilogarithms and Partition q-Series Peer-reviewed

Akishi Kato, Yuji Terashima

Communications in Mathematical Physics　338　(1)　457-481　2015/08

DOI： 10.1007/s00220-015-2323-y 　

ISSN： 0010-3616

eISSN： 1432-0916
Quiver Mutation Loops and Partition q-Series Peer-reviewed

Akishi Kato, Yuji Terashima

Communications in Mathematical Physics　336　(2)　811-830　2015/06
Publisher: Springer Science and Business Media LLC
DOI： 10.1007/s00220-014-2224-5 　

ISSN： 0010-3616

eISSN： 1432-0916
Torsion functions on moduli spaces in view of the cluster algebra Peer-reviewed

Takahiro Kitayama, Yuji Terashima

Geometriae Dedicata　175　(1)　125-143　2015/04
Publisher: Springer Science and Business Media LLC
DOI： 10.1007/s10711-014-0032-x 　

ISSN： 0046-5755

eISSN： 1572-9168
N=2 theories from cluster algebras Peer-reviewed

Y. Terashima, M. Yamazaki

Progress of Theoretical and Experimental Physics　2014　(2)　23B01　2014/02/01
Publisher: Oxford University Press (OUP)
DOI： 10.1093/ptep/ptt115 　

eISSN： 2050-3911
Semiclassical analysis of the 3d/3d relation Peer-reviewed

Yuji Terashima, Masahito Yamazaki

Physical Review D　88　(2)　2013/07/18
Publisher: American Physical Society (APS)
DOI： 10.1103/physrevd.88.026011 　

ISSN： 1550-7998

eISSN： 1550-2368
Emergent 3-Manifolds from Four Dimensional Superconformal Indices Peer-reviewed

Yuji Terashima, Masahito Yamazaki

Physical Review Letters　109　(9)　2012/08/28
Publisher: American Physical Society (APS)
DOI： 10.1103/physrevlett.109.091602 　

ISSN： 0031-9007

eISSN： 1079-7114
SL(2,R) Chern-Simons, Liouville, and gauge theory on duality walls Peer-reviewed

Yuji Terashima, Masahito Yamazaki

Journal of High Energy Physics　2011　(8)　2011/08
Publisher: Springer Science and Business Media LLC
DOI： 10.1007/jhep08(2011)135 　

eISSN： 1029-8479
Chern-Weil Construction for Twisted K-Theory Peer-reviewed

Kiyonori Gomi, Yuji Terashima

Communications in Mathematical Physics　299　(1)　225-254　2010/10

DOI： 10.1007/s00220-010-1080-1 　

ISSN： 0010-3616
Discrete Torsion Phases as Topological Actions Peer-reviewed

Kiyonori Gomi, Yuji Terashima

Communications in Mathematical Physics　287　(3)　889-901　2009/05
Publisher: Springer Science and Business Media LLC
DOI： 10.1007/s00220-009-0736-1 　

ISSN： 0010-3616

eISSN： 1432-0916
Chern-Simons variation and Deligne cohomology Peer-reviewed

Masanori Morishita, Yuji Terashima

Contemporary Mathematics　484　127-134　2009
On Poisson functions Peer-reviewed

Yuji Terashima

JOURNAL OF SYMPLECTIC GEOMETRY　6　(1)　1-7　2008/03

ISSN： 1527-5256

eISSN： 1540-2347
Geometry of polysymbols Peer-reviewed

Masanori Morishita, Yuji Terashima

MATHEMATICAL RESEARCH LETTERS　15　(1)　95-115　2008/01

ISSN： 1073-2780
ARITHMETIC TOPOLOGY AFTER HIDA THEORY Peer-reviewed

Masanori MORISHITA, Yuji TERASHIMA

Intelligence of Low Dimensional Topology 2006, Knots and Everything Series, World Scientific　213-222　2007/05
Publisher: WORLD SCIENTIFIC
DOI： 10.1142/9789812770967_0027 　
Integrable Systems, Topology, and Physics Peer-reviewed

Yuji Terashima

Contemporary Mathematics　309　291-312　2002
Publisher: American Mathematical Society
DOI： 10.1090/conm/309 　

ISSN： 0271-4132

eISSN： 1098-3627
Higher-dimensional parallel transports Peer-reviewed

Kiyonori Gomi, Yuji Terashima

Mathematical Research Letters　8　(1)　25-33　2001
Publisher: International Press of Boston
DOI： 10.4310/mrl.2001.v8.n1.a4 　

ISSN： 1073-2780

eISSN： 1945-001X
A fiber integration formula for the smooth Deligne cohomology Peer-reviewed

Kiyonori Gomi, Yuji Terashima

International Mathematics Research Notices　2000　(13)　699-699　2000
Publisher: Oxford University Press (OUP)
DOI： 10.1155/s1073792800000386 　

ISSN： 1073-7928

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Research Projects 11

Quantum Modular Forms and their Applications

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Kyushu University

2022/04/01 - 2027/03/31
Quantum Modular Forms and their Applications

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Kyushu University

2022/04/01 - 2027/03/31
二次特性類と多重対数関数の幾何

寺嶋郁二

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 基盤研究(C)

Institution: 東北大学

2021/04/01 - 2025/03/31

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トポロジー，数理物理，数論をつなげる研究を推進し，その成果として真鍋征秀氏・寺嶋靖治氏との共同研究において，ボーテックス分配関数として結び目の色つきジョーンズ多項式を与える新しい超対称ゲージ理論を構成した論文を M. Manabe, S. Terashima, Y. Terashima, The colored Jones polynomials as vortex partition functions, arXiv:2110.05662, JHEP (2021), no. 12, Paper No. 197 として公表し，学術論文誌に掲載された．また，小谷久寿氏との共同研究において，写像類群と絶対ガロア群の類似に基づいて，トポロジーにおけるOrr不変量の数論版を導入し，その性質を調べた論文が H. Kodani, Y. Terashima, Arithmetic Orr invariants of absolute Galois groups, to appear in IMRN として学術論文誌に掲載されることになった．これらはトポロジーと数理物理の橋渡しになる結果であり，さらに深めていきたい．
Geometry of secondary characteristic classes

Terashima Yuji

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

2017/04/01 - 2021/03/31

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Connecting topology to mathematical physics and number theory, we prove that specializations of cluster variables are identified with Alexander polynomials of 2-bridge knots in a joint work with Wataru Nagai, and get a result in arithmetic topology generalizing the reciprocity law about Riemann surfaces to 3-dimensional manifolds based on an analogy between foliated structures on 3-manifolds and number fields in a joint work with Junhyeong Kim, Masanori Morishita and Takeo Noda. Moreover, we introduce WRT functions, and prove that specializations of WRT functions to roots of unity are identified with WRT invariants for Seifert loops in a joint work with Hiroyuki Fuji, Kohei Iwaki and Hitoshi Murakami.
Geometry of secondary characteristic classes

Terashima Yuji

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Tokyo Institute of Technology

2013/04/01 - 2017/03/31

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We have introduced new quantities called partition q-series for quiver mutation sequences and proved fundamental properties in a joint work with Akishi Kato. One motivation is to provide a solid mathematical foundation to extract essential information of the partition function of a 3-dimensional gauge theory proposed in a joint work with Masahito Yamazaki. In particular, for quivers which are of Dynkin type or square products thereof with special mutation sequences, we have proved that partition q-series are identified with fermionic character formulas of certain modules associated with affine Lie algebras. Moreover, for reddening sequences, we have proved that a graded version of partition q-series essentially coincides with the combinatorial Donaldson-Thomas invariant.
Explicit constructions of secondary characteristic classes

TERASHIMA Yuji

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Young Scientists (B)

Institution: Tokyo Institute of Technology

2010 - 2012

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We have found a new relation of 3-dimensional gauge theory and3-dimensional hyperbolic geometry with Masahito Yamazaki in 2011. We have written a paper on the results. The paper has been published as Y. Terashima and M. Yamazaki, SL(2,R)Chern-Simons, Liouville, and Gauge Theories on Duality Walls, JHEP， 1108, 135. (2011).Following this work, with Masahito Yamazaki, we have found a new relation of indices in4-dimensional gauge theories and hyperbolic volumes of 3-dimensional hyperbolic manifolds with boundary. We have written a paper on the results. The paper has been published as Y. Terashima, M. Yamazaki, Emergent 3-manifolds from four dimensionalsuperconformal indices, Phys. Rev. Lett. ，109, (2012) 091602. This relation suggestsvarious new mathematical results.
Developing Arithmetic Topology

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Kyushu University

2007 - 2008
中興束の幾何学とその無限小としての超多様体上のホモロジーベクトル場

寺嶋郁二

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 若手研究(B)

Institution: 東京工業大学

2005 - 2008

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ホモロジーベクトル場を用いて,準ポアソン構造と捩れポアソン構造を含むより統一的な構造を幾何的に捕らえることができたので,その変形理論を展開している.特に,モジュライ空間上に自然な平坦束を得ることができた.現在,その平坦束のホロノミーの幾何的な意味を明確にすることは興味深い問題であると考えている.さらに,梶浦宏成氏との共同研究で得られたホモロジーベクトル場の変形理論の一般論と組み合わせることで,より深い情報が得られることが分かりつつある. 具体的な例として,リー群の自分自身への共役作用からくる準ポアソン構造の場合に何が起こるのかをはっきり捕らえたい.五味清紀氏との共同研究によって, C.Vafaが発見した軌道体モデルにあらわれる離散トーション位相が高次のホロノミーとして幾何的に自然に解釈されることを示すことができたので,M. Douglasらによって指摘されている離散トーション位相と非可換幾何学との関係を具体的な例について,調べている.重要な点として,離散トーション位相はもともと有限群の作用についての理論であったが,私たちの仕事によってリー群の作用を含む一般の場合についても同様の理論が展開できることになったことがある.したがって,私たちの仕事を利用して, M.Douglasらの仕事の適切な一般化を見出すことは,非可換幾何学の一つのアプローチとして興味深いと考えている.
Stady of Arithmcfc Topology

MORISHITA Masanori, TAGUCHI Yushiro, KANHARA Masato, FUJII Michihiko, TERASHMA Yuji, KOHNO Toshtake

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: KYUSHU UNIVERCITY

2005 - 2006

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Based on analogies between knots and primes, I investigated the connection and interaction between knot theory and number theory. In this year, my concern has been focused on the following topics : I showed the analogy between the structure of the deformation space of hyperbolise structures on a knot complement and the deformation space of p-adic Galvis representations of a prime group. Based on this analogy, I discussed some analogous features between the variation of Sh_2(C) Chern-Simons. in variants and p-adre modulus L-function. In particular, I gave an interpretation of the Chern-Simons invariant in terms of Deligne cohomology, Jointly with Y.Terashima. I talked about Dnr results in the international conferees hold at Hiroshima Univ.(2006,July), Max Plank Inst.(2006,Sept) and Kyoto Un.(2006,Dec)
Various inbariants appearing in low-dimensional topology

MURAKAMI Hitoshi, TERASHIMA Yuji, ISHIKAWA Masaharu, KITANO Teruaki, USHIJIMA Akira, ENDO Hisaaki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Tokyo Institute of Technorogy

2003 - 2006

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I am working on the volume conjecture for knots and its various generalizations. The volume conjecture states that a certain limit of the colored Jones polynomial of a knot would determine the volume of the complement of the knot. More precisely, we replace the paramaeter of the colored Jones polynomial of 'color' N with exp(2Pi^*I/V) and then consider the limit where N goes to the infinity. So far I generalized the volume conjecture as follows : If we replace the parameter with exp(a/N), change a variously, and take the limit, then we would have not only the volume of the knot complement but also the volume and the Chern-Simons invariant of the three-manifold obtained from the knot by Dehn surgery. In this academic year, I studied not only the limit but also several coefficients of the asymptotic expansion of the logarithm of the colored Jones polynomial with respect to large N. The volume conjecture is equivalent to saying that the coefficient of N would determine the volume and the Chern-Simons invariant. As a result of a joint work with S.Gukuv, we proposed the following new conjecture : 1.The coefficient of log N would be determined by the dimension of the cohomoloty group twisted by a representation of the knot group at SL(2, C). 2.The constant term would be determined by the Reidemeister torsion corresponding to a representation of the knot group at SL(2, C). The conjecture above gives a new aspect to the volume conjecture and its generalizations. Moreover, we have confirmed by computer caluculations that this conjecture is true for some knots.
非線形シグマ模型における位相不変量とアフィン・リー環の表現論

寺嶋郁二

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 特別研究員奨励費

Institution: 東京大学

2001 - 2004

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ドリーニュ・コホモロジーのコチェインのレベルの転入写像を与えられた多様体が境界をもたない場合に適用し、高次のホロノミーのホロノミーの概念を得て、チーガー・サイモンズの微分指標群とドリーニュ・コホモロジーの同型を具体的に与えることができた。この同型は、古典的なホロノミーが一次元の微分指標の典型的な例であるというCheeger-Simonsが指摘した観点を高次元に拡張するものとして考えられる。古典的な場合との違いは、局所的に微分方程式を解くことで高次元のホロノミーを得ることが出来ない点にある。つまり、古典的な場合は閉曲線を小さな閉区間に分け局所的に微分方程式を解くことでホロノミーを得ることができたが、一般の多様体を細かく分ける方法は多様で、しかもその過程で様々な多様体が'局所的'に現れる。それらの'局所的'な対象を同時に扱うための道具としてドリーニュ・コチェインの転入写像を導入した。構成におけるアイデアは、単体分割の最高次数の各単体についての局所的な寄与を足しあげて得られる微分形式の積分の概念を拡張し、各フラッグについての寄与を足しあげ、それと高次のホモトピー作用素を組み合わせることにあり、それを精密化することで滑らかな空間への拡張を得た。これらの結果をContemp. Math.に掲載した。これに関連して、次数3のドリーニュ・コホモロジー類に対応する幾何的な対象である接続つきの主圏束の'無限小'として考えられるホモロジカル関数の変形理論を用いて準ポアソン構造と捩れポアソン構造を統一的に解釈できた。これらの結果を研究集会『接触構造、特異点と関連分野』で発表した。

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