| Project/Area Number |
16540352
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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 | Kisarazu National College of Technology |
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Principal Investigator |
KAMATA Masaru Kisarazu National College of Technology, Natural Science Education, Professor (10169609)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAMURA Atsushi Kitasato University, Department of Physics, School of Science, Lecturer (90245415)
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| Project Period (FY) |
2004 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,740,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | Yang-Mills fields / Self-dual / ADHMN construction / q-analogue / Nahm equation / Integrable system / Singularity confinement test / Algebraic entropy / 簡約 / スペクトル曲線 |
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| Research Abstract |
In mathematical physics, the ADHM (ADHMN) construction is a well-known method to construct most generally the (anti-) selfdual Yang-Mills fields in the four-dimensional Euclidean space R^4 (R×R^3). The head investigator and the investigator presented, in a previous paper, a q-analogue of the ADHMN construction based on the countably infinite dimensional Hilbert space l^2 defined at a discrete q-interval I_q:={±1/2, ±q/2, ±^2/2, ., ±q^n/2, .} with a real parameter q. Further, as a q-analogue of the Nahm equations which describe the multi-monopole configurations in R×R^3, the following q-discretized Nahm equations (or q-Nahm equations in short) are obtained:
D_qT_j(z)=(1/2)ε_{jkl}(T_k(qz)T_l(z)-T_l(qz)T:_k(z)), j,k,l=1,2,3
where D_q is the q-difference operator.
The following have been studied in this project in the above term :
1. The Euler-top type reduction of the q-Nahm equations yields a system of difference equations which pass the singularity confinement test.
2. A Toda-Flaschka type reduction of the q-Nahm equations yields a system of difference equations with at least one conserved quantity.
3. A q-analogue of the caloron solution in S^l×R^3 is obtained through the ADHM construction.
4. Relations between the Nahm equations and the Euler top equations are studied.
5. Singular solutions derived from the known solutions to the CBS equations are investigated.
6. An extended KP hierarchy with the Riemann-Liouville differential-integral operators of the order 1/3 is examined.
7. A preliminary calculation of the algebraic entropy to the q-Nahm equations is given.
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