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Crystal melting and wall crossing phenomena

Yamazaki, Masahito (2011) Crystal melting and wall crossing phenomena. International Journal of Modern Physics A, 26 (7-8). pp. 1097-1228. ISSN 0217-751X.

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This paper summarizes recent developments in the theory of Bogomol'nyi–Prasad–Sommerfield (BPS) state counting and the wall crossing phenomena, emphasizing in particular the role of the statistical mechanical model of crystal melting. This paper is divided into two parts, which are closely related to each other. In the first part, we discuss the statistical mechanical model of crystal melting counting BPS states. Each of the BPS states contributing to the BPS index is in one-to-one correspondence with a configuration of a molten crystal, and the statistical partition function of the melting crystal gives the BPS partition function. We also show that smooth geometry of the Calabi–Yau manifold emerges in the thermodynamic limit of the crystal. This suggests a remarkable interpretation that an atom in the crystal is a discretization of the classical geometry, giving an important clue as such to the geometry at the Planck scale. In the second part, we discuss the wall crossing phenomena. Wall crossing phenomena states that the BPS index depends on the value of the moduli of the Calabi–Yau manifold, and jumps along real codimension one subspaces in the moduli space. We show that by using type IIA/M-theory duality, we can provide a simple and an intuitive derivation of the wall crossing phenomena, furthermore clarifying the connection with the topological string theory. This derivation is consistent with another derivation from the wall crossing formula, motivated by multicentered BPS extremal black holes. We also explain the representation of the wall crossing phenomena in terms of crystal melting, and the generalization of the counting problem and the wall crossing to the open BPS invariants.

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Additional Information:© 2011 World Scientific Publishing Company. Received 24 February 2010; Revised 12 December 2010. This research project would not have been realized by me without the help, support and inspiration from many people. First and foremost, it is a pleasure to express my gratitude to Prof. Hirosi Ooguri. He has been my adviser during the last two years of my Ph.D. course at IPMU and Caltech, and this paper owes much to the discussions and collaborations with him. I have learnt a lot from him, and he has offered me invaluable assistance, guidance and support. I owe my deepest gratitude to my adviser Prof. Tsutomu Yanagida. During my stay at the University of Tokyo, I have often enjoyed the discussion with him about physics, which is always inspiring and suggestive to me. I also had a wonderful opportunity to collaborate with him. I am also deeply indebted to Prof. Tohru Eguchi, who is my mentor since my Master course study at the University of Tokyo. The research on brane tilings and quiver gauge theories, which I have undertaken under his supervision, provides the basic ground, which has led me to the study described in this paper. Special thanks go to my collaborators: Mina Aganagic, Kentaro Nagao and Cumrun Vafa. It has been a wonderful and an exciting experience to collaborate with you all. I have also obtained valuable feedback from the discussion with Alexei Borodin, Tudor Dimofte, Lotte Hollands, Sergei Gukov, Daniel Krefl, Andrei Okounkov, Kazushi Ueda, Jaewon Song, Piotr Su lkowski and Yukinobu Toda. I would also like to express my gratitude to the Ph.D. paper committee, especially Yoichi Kazama, for criticism and comments on the manuscript, which is partly incorporated into this final version. I am indebted to my colleagues at the University of Tokyo, both at the Department of Physics and at the Institute for the Physics and Mathematics of the Universe. I would also like to thank the high energy theory group at Caltech for the hospitality, where I enjoyed my one year stay in total. I would also like to thank the secretaries, Mami Hara at Tokyo and Carol Silberstein at Caltech. The contents in this paper are presented at numerous universities and workshops that I have attended. I convey my thanks to all the participants for their interest, questions and feedback, which greatly serve to improve this paper. During my Ph.D study, I have been supported by the JSPS fellowships for Young Scientists, by the Global COE Program for Physical Sciences Frontier at the University of Tokyo funded by MEXT of Japan, by DOE grant DE-FG03- 92-ER40701, and by the World Premier International Research Center Initiative of MEXT of Japan. I would like thank all these funding agencies for their support. Last but not least, I wish to thank my parents for their understanding, love and support.
Funding AgencyGrant Number
Japan Society for the Promotion of Science (JSPS)UNSPECIFIED
Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT)UNSPECIFIED
Department of Energy (DOE)DE-FG03-92-ER40701
World Premier International Research Center InitiativeUNSPECIFIED
Subject Keywords:Crystal melting; wall crossing phenomena; topological string theory
Issue or Number:7-8
Classification Code:PACS: 11.25.Uv, 11.25.Yb, 11.25.Mj
Record Number:CaltechAUTHORS:20110425-101326675
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:23446
Deposited By: Tony Diaz
Deposited On:27 Apr 2011 16:20
Last Modified:03 Oct 2019 02:46

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