2011 Fiscal Year Final Research Report
Study of topological condensed matter problems by random matrices
| Project/Area Number |
21540380
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Mathematical physics/Fundamental condensed matter physics
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| Research Institution | The University of Tokyo |
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Principal Investigator |
HIKAMI Shinobu 東京大学, 大学院・総合文化研究科, 教授 (30093298)
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| Project Period (FY) |
2009 – 2011
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| Keywords | 数理物理 / メゾスコピック系 / リーマン面 / トポロジー |
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| Research Abstract |
The topological invariance of the moduli space of Riemann surface is shown to be obtained from the Gaussian random matrix model with an external source by the tuning of external source. The invariance is classified by a parameter p, which characterizes the degeneracy of the external source. The invariance is related to the p-spin curves on the Riemann surface. This topological singularity is related the bifurcation of the growth of crystals. The parameter p is extended to the negative values(p=-1,-2,...) and it represents level k SL(2, R)/U(1) Wess-Zumino-Witten term. The case of p=-2 corresponds to the unitary matrix model, which has a phase transition. The strong and weak expansion are examined by Gaussian random matrix theory. This transition is studied in the relation of the phase transitions in the condensed matter problems. The time dependence of the random matrix model is studied by the reduction to 2-matrix model and the algebraic structure is investigated in details. The algebraic structure is found to be the associated W-algebra. The structure represents the deviation from N=2 supersymmetric minimal model and the change of the Ramond-Ramond term for the time dependence is studied.
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