2004 Fiscal Year Final Research Report Summary
Algebraic and geometric aspects of string dualities
| Project/Area Number |
13640264
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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 |
素粒子・核・宇宙線
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| Research Institution | The University of Tokyo |
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Principal Investigator |
KATO Akishi The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (10211848)
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| Project Period (FY) |
2001 – 2004
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| Keywords | quantum field theory / string theory / supersymmetric gauge theory / duality / renormalization group / derived category / rational elliptic surface / moduli space |
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| Research Abstract |
The renormalization of quantum field theories are commonly phased as "the procedure to extract finite results from divergent quntities by subtracting infinity." This obscures why one can get a reliable answer avoiding arbitrariness. Cannes and Kreimer proposed a Hopf algebra point of view to the renormalization. In their paper, however, it was not completely clear which part is assumption and which part is logical consequence. As a part of our project of understanding dualities algebraically, I started to clarify the relationship between Hopf algebra and theory of renormalization. In particular, I studied the renormalizability and the Birkhoff decomposition in detail using a toy model.
Discovery of string dualities enabled us to "geometrize" the various phenomena in physics, and provided a tool to analyze the dynamics beyond the reach of perturbative method. In particular, new type of fixed points were discovered in six dimensional N=(1,0) supersymmetric gauge theories with E8 global symmetry. F-theoretic interpretation of this critical point is the vanishing locus of codimension one rational elliptic surface in Calabi-Yau threefold. Conjecturally elementary excitations called "E-strings" will play an important role in the critical point. As has been successful in Seiberg-Witten theory, renormalization group flow can be seen as the deformation family of elliptic curves fibered over a moduli space. The duality group of the quantum field theory acts not only as covering transformations associated with the fibration, but also the autoequivalence of the derived categories on the surface. I investigated the relationship between the geometry of rational elliptic surfaces and the dynamics of the supersymmetric gauge theories, especially the partition functions of the topological gauge theories (E-strings). This work was done with Professors H.Awata (Nagoya), S.Kondo (Nagoya), Y.Saito (Tokyo), Y.Shimizu (ICU) and A.Tsuchiya (Nagoya).
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Research Products
(8 results)
