2004 Fiscal Year Final Research Report Summary
Study on the anomaly of Spin^q manifolds
| Project/Area Number |
14540062
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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 |
Geometry
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| Research Institution | Saitama University |
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Principal Investigator |
NAGASE Masayoshi Saitama University, Dept.of Math., Professor, 理学部, 教授 (30175509)
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| Co-Investigator(Kenkyū-buntansha) |
MIZUTANI Tadayoshi Saitama University, Dept.of Math., Professor, 理学部, 教授 (20080492)
SAKAMOTO Kunio Saitama Univ., Dept.of Math., Professor, 理学部, 教授 (70089829)
FUKUI Toshizumi Saitama Univ., Dept.of Math., Professor, 理学部, 教授 (90218892)
SAKAI Fumio Saitama Univ., Dept.of Math., Professor, 理学部, 教授 (40036596)
SHIMOKAWA Koya Saitama Univ., Dept.of Math., Associate Professor, 理学部, 助教授 (60312633)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Spin / Spin^q / twistor space / chiral anomaly / Dirac operator |
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| Research Abstract |
His previous study says that a Spin^q manifold (M,g^M] possesses a canonical CP^1-fibration and its total space (Z,g^z=π^*g^M+g^<C_<P^1>>) called a twistor space has a canonical Spin structure. The structure induces the Dirac operator 【∂!/】 on Z.
(1)The definition of the infinitesimal chiral anomaly and what should be studied : First, its infinitesimal variation δ_X【∂!/】 in the X-direction, where X is a cross-section of a certain adjoint bundle, and its anomaly log det δ_X【∂!/】≡-∂/∂s|_<s=0>^1/(2Γ(s)) f^∞_0 t^s STr(δ_X【∂!/】【∂!/】e^<-t【∂!/】^2>) dt were introduced (by the investigator) from the mathematician's viewpoint. After the analogy of the physical twistor theory and the creating theory of the universe, he considered the operation of collapsing each fiber into one point (returning to the pre-universe), i.e., the operation of taking the adiabatic limit, to produce its essential part denoted lim_<ε→0> log det δ_X【∂!/】_ε. To investigate the limit was the main purpose.
(2)A closer investigation into the Getzler transformation : The so-called Getzler transformation G_ε(【∂!/】_ε)^2 is a useful tool for the study. He showed it very effective to consider it as a composition of two kinds of transformations. By investigating them closely, the study introduced in (1)was quite improved.
(3)Synchronous connection and the Gilkey theory : He introduced a concept of synchronous connection denoted ▽^<g^Z>【symmetry】, which is a little bit simpler than the Levi-Civita one. Accordingly, we define synchronous geodesies, coordinates, etc., and come to the study some invariant polynomials of the curvatures coefficients of ▽^<g^Z>【symmetry】. Unfortunately, it is not yet finished and, to finish it, further invesatigation into typical Spin^q manifolds will be needed.
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