# FY2015 Annual Report

**Mathematical and Theoretical Physics Unit
Professor Shinobu Hikami**

## Abstract

Mathematical and theoretical physics unit studied the subjects related to random matrix theory and conformal field theory. The intersection numbers of the moduli space of spin curves and the conformal bootstrap method for three dimensions are studied. Random matrix theory is applied to biological systems.

## 1. Staff

- Dr. Shinobu Hikami, Professor
- Dr. Hirohiko Shimada, Post Doctoral Scholar
- Dr. Tomoki Tokuda, Post Doctoral Scholar
- Dr. Ayumi Kikkawa, Research Assistant (part time)
- Dr. Takayuki Oda, Staff Scientist (part time)
- Dr. Momoka Higa, Technician
- Mr. Takuro Tamashiro, Technicien
- Dr. Satsuki Oda, visiting researcher
- Ms. Miwako Tokuda, Research Unit Administrator

## 2. Collaborations

### 2.1 Theme: Random matrix theory

- Type of collaboration: Joint research
- Researchers:
- Professor E. Brézin, Ecole Normale Supérieure

## 3. Activities and Findings

### 3.1 Random matrix theory

The random matrix theory with an external source was investigated for multicritical points. The open boundary of Riemann surface is studied by Kontsevich-Penner model. The intersection numbers for non-orientable surfaces were studied by the random matrix theory of Lie algebra of *O(n)* and *Sp(n)*. The intersection numbers of moduli space of *p*-spin curves was evaluated, for one marked point, up to genus *9,* and the relation between *p=–1* and *p* infinity is derived. The singularities in the random matrix theory with an external source are classified.

### 3.2 Conformal field theory

The fractal dimensions of polymer chains and high-temperature graphs in the Ising model both in three dimension are determined using the conformal bootstrap applied for the continuation of the *O*(*N*) models from *N*=1 (Ising model) to *N*=0 (polymer). The unitarity bound below *N*=1 of the scaling dimension for the the *O*(*N*) -symmetric-tensor develops a kink as a function of the fundamental field as in the case of the energy operator dimension in the Ising model. Although this kink structure becomes less pronounced as *N* tends to zero, an emerging asymmetric minimum in the current central charge *CJ* can be used to locate the CFT. It is pointed out that certain level degeneracies at the *O*(*N*) CFT should induce these singular shapes of the unitarity bounds. As an application to the quantum and classical spin systems, we also predict critical exponents associated with the *N*=1 supersymmetry, which could be relevant for locating the corresponding fixed point in the phase diagram.

### 3.3 Random matrix approach for protein-protein interaction network

The proteins interact with each other in cells and such interacting networks manifest the biological response to the environments. The study of protein-protein interaction (PPI) network is very important in order to understand the functions and the dynamics of the biological systems. We apply random matrix theory (RMT) to the PPI networks in various species including yeast, human and Arabidopsis. We also compare the human PPI networks between the cancer proteins and non-cancer proteins.

From the yeast PPI data, we select 1,951 proteins and a total of 43,755 interactions and obtain the nearest neighbor level spacing distribution *P(s)*. From human PPI data, we plot the nearest neighbor level spacing distribution *P(s)* with 138,327 interactions of 14,397 proteins. These data have been downloaded from the BioGRID database (http://thebiogrid.org). In both cases, *P(s)* become Wigner-Dyson distribution of GOE random matrix theory.

In cancer cells, the PPI network has been constructed for the entire human genome using an orthology-based computational inference method. (Jonsson and Bates, 2006) From this dataset, we plot the *P(s)* with a total of 7,014 interactions between 4,813 proteins. In this case, the Poisson distribution is observed. The small *s* behavior of the level spacing distribution is very important to distinguish the network topology of the system. On the other hand, it has been experimentally shown that the cancer hub proteins interact with much more proteins compared to normal proteins. The PPI network is thus predicted to be separated into some sub-networks in cancer cells. In such cases, the nearest neighbor level spacing distribution *P(s)* should behave as the Poisson distribution.

The cancer PPI data that we have used in this study is highly dependent on the computational inference methods. In order to analyze the topological property of the cancer PPI networks in details, much more studies are required and we consider the RMT gives an effective clue to such investigations.

## 4. Publications

### 4.1 Journals

- E. Brezin and S. Hikami, Random Matrix, Singularities and Open/Close Intersection Numbers, J. Phys. A Math. Theor. 48 (2015) 475201.
- H. Shimada and S. Hikami, Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D conformal bootstrap, arXiv:1509.04039, 2015
- Satsuki Oda, Nobuchika Okada, and Dai-suke Takahashi, Classically conformal (1)’ extended standard model and Higgs vacuum stability Phys. Rev. D 92, 015026
- T. Tokuda et al., Multiple co-clustering based on nonparametric mixture models with heterogeneous marginal distributions, arXive:1510.06138, 2015

### 4.2 Books and other one-time publications

Nothing to report

### 4.3 Oral and Poster Presentations

- S. Hikami, p-spin curve and Euler characteristics, MCM2015Autumn, Univ. of Tokyo, Oct. 29, 2015.
- H. Shimada, One parameter families of CFT via the Virasoro/SL(2,C) decomposition and the unitarity bound, JPS 2015 Autumn Meeting, Kansai university, Sep. 14, 2015.
- H. Shimada, Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D CFT, MCM2015Autumn, Univ. of Tokyo, Oct. 30, 2015.
- H. Shimada, One-parameter fractals in 3D non-unitary CFT, the 10th Asian Winter School on Strings, Particles and Cosmology, OIST, Jan. 9, 2016.
- A. Kikkawa, Random matrix analysis for molecular networks in biological systems, RMT2015, OIST C209, Nov.5, 2015.
- T. Tokuda, J. Yoshimoto, and K. Doya, Multiple clustering in nonparametric mixture models based on co-clustering views, ICML workshop on Features and Structures, Lille Grand Palais, Lille, France, July 10, 2015.
- T. Tokuda, J. Yoshimoto, Y. Shimizu, and K. Doya, Multiple clustering based on co-clustering views, IJCNN 2015 workshop on Advances in Learning from/with Multiple Learners, Killarney Convention Centre, Killarney, Ireland, July 16, 2015.
- T. Tokuda, J. Yoshimoto, Y. Shimizu and K. Doya, Multiple co-clustering based on nonparametric mixture models with heterogeneous marginal distributions, International Meeting on High-dimensional data drive science, Meruparuku, Kyoto, Dec. 16, 2015.

## 5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

## 6. Meetings and Events

### 6.1 MCM2015Autumn

- Thursday, October 29 and Friday, October 30, 2015
- Venue:Komaba, University of Tokyo, Math. Dep. Room 002.

### 6.2 RMT2015: Random matrix theory from fundamental mathematics to biological applications

- Date: Monday, November 2nd - Friday, November 6th, 2015
- Venue: OIST Campus C209

### 6.3 Workshop on "Moduli spaces of abelian varieties and curves, and related analysis"

- Dates: December 15, 2015 - December 18, 2015
- Venue : Graduate School of Mathematical Sciences, the Univ. of Tokyo

### 6.4 Seminar:

- Date: Wednesday, May 27th, 2015
- Time: 13:00-14:00
- Venue: Lab.1, C016Speaker: Dr. Serguei Brazovskii from CNRS & University Paris-Sud, Orsay, France
- Title: Solitons in low dimensional electronic systems of synthetic conductors.

### 6.5 Seminar:

- Date: Wednesday, June 3rd, 2015
- Time: 13:00-14:00
- Venue: Lab.1, C016
- Speaker: Dr. Natasha Kirova from CNRS & University Paris-Sud, Orsay, France
- Title: Electronic ferroelectricity in carbon-based systems: from reality of organic conductors to promises of polymers

### 6.6 Seminar:

- Date: July24, 2015
- Venue: B250, OIST
- Speaker: Dr. Tomoki Tokuda (Hikami Unit, OIST)
- Title: Multiple clustering for high-dimensional data

### 6.7 Seminar:

- Date: Friday, November 20, 2015
- Time: 11:00-12:00
- Venue: Lab.1, C016
- Speaker: Dr. Tsunehide Kuroki (Kagawa Kosen)
- Title: Spontaneous supersymmetry breaking in matrix model and superstring theory

### 6.8 Seminar:

- Date: Friday, December 25th, 2015
- Time: 11:00 - 12:00
- Venue: Lab1, C015
- Speaker: Prof. Yasutomo J. Uemura (Columbia University)
- Title: BEC to BCS crossover picture for high-Tc cuprate and unconventional superconductors, and pairing due to comparable spin-charge energy scales: message from muons and neutrons

## 7. Other

Nothing to report.