# FY2016 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

- Prof. Shinobu Hikami, Professor
- Prof. Takayuki Oda, Staff Scientist
- Dr. Ayumi Kikkawa, Research Assistant (part time)
- Dr. Hirohiko Shimada, Postdoctoral Resaercher
- Dr. Tomoki Tokuda, Postdoctoral Resaercher
- Dr. Momoka Higa, Technician
- Mr. Takuro Tamashiro, Technician
- Mr. Adrian David, PhD Student
- Dr. Riki Kobayashi, (Visiting Researcher)
- Dr. Satsuki Oda, (Visiting Researcher)
- Ms. Shiho Saito, Research Unit Administrator
- Ms. Miwako Tokuda, Research Unit Administrator

## 2. Collaborations

### 2.1 Moduli space and matrix model

- Type of collaboration: Joint collaboration with University of Tokyo
- Researcher:
- Professor Nariya Kawazumi, University of Tokyo

### 2.2 Random matrix theory

- Type of collaboration: Joint research
- Researcher:
- Professor Edouard Brezin (lpt, ENS, Paris)

## 3. Activities and Findings

### 3.1 Random matrix models and intersection theory

The intersection numbers of the moduli space of p spin curves have been studied through the random matrix theory with an external source. The topological invariants of Gromow-Witten are explicitly obtained for P^{1 }case. When the Riemann surface has a boundary, i.e. hole, the intersection numbers are obtained from Kontsevich-Penner matrix model. We generalized this open intersection numbers for p-spin curves. We published a book for these studies (Brezin and Hikami, SpringerBrief, Mathematical Physics, Vol. 19 (2016)).

This random matrix theory with an external source is also related to Klein surface. We had a workshop MCM2016 for higher Teichmuller space, in which we discussed super Riemann surface, and moduli space of matroid.

### 3.2 Conformal Field Theory

We studied critical phenomena in d>2 by the conformal bootstrap, which enables us to explore the fixed point of the renormalization group (RG) i.e. the conformal field theory from certain consistency conditions, the most important being the crossing symmetry and the unitarity. We gave precise estimates for the fractal dimensions of critical geometric objects in d=3, such as the polymer [1.7016(36)] and the Ising graph [1.7346(5)] (giving the world-record precision) obtained respectively in the n=0 limit (as done in RG by de Gennes) and the rank-2 tensor sector at n=1 of the O(n) model. Non-integer values of n are also studied, for which the results may be viewed as a generalization of the continuous family in the Beffara's theorem of the Schram-Loewner evolution in d=2. We found both weak and strong violations of the unitarity happen in the interval 0<n<1, therefore giving an excellent target for the further study deepening our understanding on the fundamental implications of the non-unitarity (non-positive geometry). We also had useful discussions in the focus week of the conformal bootstrap workshop held in GGI, Florence. This project was supported by JSPS KAKENHI No. 16K05491 Grant-in-Aid for Scientific Research (C) and also by No. 15K13540 Grant-in-Aid for Challenging Exploratory Research.

### 3.3 Big Data and Earthquakes

Identifying precursors to large earthquakes has been a difficult, but important challenge both in science and in our society. As the available data have increased drastically in the past decades, there have been steady accumulations of tantalizing evidences that this should be possible as communicated to us through the enlightening discussions with Tetsuo Yamaguchi (Kyushu University) and Takahiro Hatano (Earthquake Research Institute). We studied two directions both using the big data: the one is based on detecting the critical increase of the sensitivity to the tidal-triggering (p-value), and the other is based on machine-learning the inter-time distribution of the low frequency earthquakes (LFEs). The p-value is calculated to introduce a hypothetical 2-dimensional incompressible fluid, in which the power-law of the fully-developed turbulence is observed in a narrow range of the velocity power spectrum; the precursor is found several months before a few large earthquakes as a tiny vortex which may grow via the inverse energy cascade. On the other hand, we focus on the inter-time distribution of the LFEs among the various manifestations of the slow earthquakes, which have recently garnered much attention in the hope of understanding the mechanisms of earthquakes; four homogeneous classes of the LFEs in northern Japan are identified via the cluster analyses in machine-learning the JMA catalogue. Remarkably, the shortest inter-time class (median 23 seconds) is found to show both a significantly-lower seismicity about four months before and a complete quiescence (p-value = 0.0002) of 30 days before the 2011 Great Tohoku Earthquake.

### 3.4 Random Matrix Analysis for Gene Interaction Networks

A cell can be considered as a complex universe of various interacting molecules including proteins, RNA and many other transcripts. There are more than 10k proteins in each human cell at the same time. Every cell contains the DNA in their nuclei coding more than 30k genes.

Recently, this huge transcript networks can be observed experimentally using the high-throughput equipment such as microarrays or the second-generation sequencing. The transcript networks manifest the cellular behavior, so the difference between the normal cells and disease cells such as cancers has been investigated deeply. There are also huge computational studies which infer interactions between the members of the transcript networks.

We study gene interaction networks in various human cancer cells with the random matrix theory. We apply the random matrix theory to the computationally inferred gene interaction networks and found the universal behaviors. We observed the Wigner’s surmise is valid in the eigenvalue density function of interaction matrices of dense gene interaction networks. The distribution of nearest neighbor level spacing becomes the Wigner distribution when the network size is large, and it behaves as the Poisson distribution when the network size is small. We expect that the random matrix theory provides an effective analytical method for investigating the huge interaction networks of the various transcripts in cancer cells.

## 4. Publications

### 4.1 Journals

- Arindam Das, Satsuki Oda, Nobuchika Okada, Dai-suke Takahashi, “
*Classically conformal U(1)′extended standard model, electroweak vacuum stability, and LHC Run-2 bounds*”, Phys. Rev. D 93, 115038 (2016) - Hirohiko Shimada, Shinobu Hikami, “
*Fractal Dimensions of Self-Avoiding Walks and Ising High-Temperature Graphs in 3D Conformal Bootstrap*”, J. Stat. Phys. 165, 1006–1035 (2016)

### 4.2 Books and other one-time publications

- Edouard Brézin, Shinobu Hikami, “
*Random matrix theory with an external source*”, SpringerBriefs in Mathematical Physics, Springer Singapore, ISBN: 978-981-10-3316-2

### 4.3 Oral and Poster Presentations

- Hirohiko Shimada, Shinobu Hikami, “
*Fractal dimension of the polymers*”, Workshop on Conformal Field Theories and Renormalization Group Flows in Dimensions d>2, The Galileo Galilei Institute for Theoretical Physics (GGI), Florence, Italy, 2016.06.08 - Arindam Das, Satsuki Oda, Nobuchika Okada, Dai-suke Takahashi, “
*LHC Run-2 bounds on the Z’ boson mass in classically conformal U(1)’ extended SM with electroweak vacuum stability*”, The 24th International Conference on Supersymmetry and Unification of Fundamental Interactions (SUSY 2016), University of Melbourne, Australia, 2016.07.04 - Tomoki Tokuda, “
*Multiple co-clustering and its application*”, RIKEN-Osaka-OIST joint workshop Big Waves of Theoretical Science in Okinawa 2016, OIST, 2016.07.08 - Shinobu Hikami, “
*Random matrix models and applications*”, RIKEN-Osaka-OIST joint workshop Big Waves of Theoretical Science in Okinawa 2016, OIST, 2016.07.09 - Hirohiko Shimada, Shinobu Hikami, “
*Polymer dimension from fractal trajectories in 3D conformal field theory*”, RIKEN-Osaka-OIST joint workshop Big Waves of Theoretical Science in Okinawa 2016, OIST, 2016.07.09 - Ayumi Kikkawa, “
*Random matrix analysis for gene co-expression experiments in cancer cells*”, RIKEN-Osaka-OIST joint workshop Big Waves of Theoretical Science in Okinawa 2016, OIST, 2016.07.09 - Shinobu Hikami, “
*Matrix models on Riemann surface*”, Japan-Russia Working Seminar 2016, OIST, 2016.07.22 - Shinobu Hikami, “
*Intersection numbers of the moduli space of p-spin curves from matrix models*”, Tsuda College–OIST Joint Workshop Calabi-Yau Manifolds: Arithmetic, Geometry and Physics, OIST, 2016.08.03 - R Kobayashi, H Tanida, M Sera, M Frontzek, M Souda, M Masuda, H Yoshizawa, N Aso, K Kaneko, S Wakimoto, Y Uwatoko, S Hikami, “
*Neutron diffraction Study of Rh-doping-induced phase in CeRu2Al10*”, The Physical Society of Japan 2016 Autumn Meeting, Kanazawa University, 2016.09.15 - Satsuki Oda, Nobuchika Okada, Dai-suke Takahashi, “
*Drak matter in the classically conformal U(1)' extended standard model*”, The Physical Society of Japan 2016 Autumn Meeting, University of Miyazaki, 2016.09.21 - Hirohiko Shimada, “
*Hierarchies of Conformal Amplitudes*”, Workshop on Moduli space, conformal field theory and matrix models (MCM2016), OIST, 2016.10.27 - Tomoki Tokuda, “
*Statistical test for detecting community structure in real-valued edge-weighted graphs*” ACML Workshop on Learning on Big Data, Hamilton, New Zealand, 2016.11.16 - Shinobu Hikami, “
*Random matrix theory with an external source and moduli space*”, Kanazawa University, 2016.12.07 - Satsuki Oda, Nobuchika Okada, Dai-suke Takahashi, “
*Right-handed neutrino DM in the classically conformal U(1)’ extended SM”*, KEK Theory Meeting on Particle Physics Phenomenology (KEK-PH2017), 2017.02.15, KEK, Tsukuba - Satsuki Oda, Nobuchika Okada, Digesh Raut, Dai-suke Takahashi, “
*Non-minimal quartic inflation in classically conformal U(1)' extended SM”*, The 3rd Toyama International Workshop on "Higgs as a Probe of New Physics (HPNP2017), University of Toyama, 2017.03.02 - Hirohiko Shimada, “
*An SL(2, Z) Hierarchy of Conformal Amplitudes*”, JPS 72nd Annual Meeting (2017), Osaka University, 2017.03.20 - Takayuki Oda, “
*How to get non-trivial cohomology classes in Shimura varieties, a case study*”, Japan-U.S. Mathematics Institute (JAMI) 2017 Conference, Johns Hopkins University, Baltimore, USA, 2017.03.24 - Satsuki Oda, Dai-suke Takahashi, “
*Right-handed neutrino dark matter in the classically conformal U(1)’ extended Standard Model*”, University of Alabama, Tuscaloosa, Alabama, USA, 2017.03.29

## 5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

## 6. Meetings and Events

**6.1 Symposium: Geometry of Quadratic Differentials and Related Topics**

Quadratic differentials, meromorphic tensor fields on Riemann surfaces of some special type, have been studied classically in hyperbolic geometry and complex analysis. Recently it has been clarified that low-dimensional topology including hyperbolic geometry of surfaces is closely related to mathematical physics and integrable systems. This can be illustrated by the topological recursion by Eynard and Orantin generalizing computation of the Weil-Petersson volume of the moduli space of Riemann surfaces, whose influence one can find in various research areas. Accordingly quadratic differentials are playing some important roles also in mathematical physics and integrable systems. In this workshop, we will read intensively some research papers on the geometry of quadratic differentials and related topics, and will have some talks on original researches. This workshop is supported by a joint project between the OIST and the University of Tokyo.

- Organizers: Y. Tadokoro (Kisarazu), T. Sakasai (Univ. Tokyo) and N. Kawasumi (Univ. Tokyo)
- Date: January 11-13, 2017
- Venue: University of Tokyo, Komaba, Mathematical Science Building #002 Info (in Japanese)

**6.2 Pre-Conference MCM2016 at Tokyo**

- Date: October 20-22, 2016
- Venue: University of Tokyo, Komaba, Mathematical Science Building
- Organizers: Nariya Kawazumi(Univ. Tokyo), Shinobu Hikami(OIST)
- Program & Abstracts

**6.3 Workshop: Moduli space, conformal field theory and matrix models; MCM2016**

- Date: October 24-27, 2016
- Venue: C209, OIST
- Organizers: Nariya Kawazumi(Univ. Tokyo), Shinobu Hikami(OIST)
- Program & Abstracts
- Group Photo

**6.4 TSUDA COLLEGE AND OIST JOINT WORKSHOP ONCALABI–YAU VARIETIES:ARITHMETIC, GEOMETRY AND PHYSICS **

- Date: August 1–3, 2016
- Venue: August 1–2 at Tsuda College Room 7309, August 3 at Tokyo University Komaba Campus Room 117
- Organizers: Noriko Yui (Queen’s University) and Takayuki Oda (OIST)
- Program (PDF) & Abstracts (PDF)

**6.5 "OIST-iTHES-CTSR 2016" International Workshop**

- Date: Friday 8 - Monday 11 July, 2016
- Venue: Fri 8th (B250), Sat 9th (C209), Mon 11th (B250)
- Organizers: RIKEN iTHES, OSAKA CTSR, and OIST (Theoretical Units)

**6.6 Seminar**

- Date: Thursday, July 7, 2016
- Time: 15:00 - 16:00
- Venue: C209, Center Building, OIST
- Speaker: Prof. Jean Zinn-Justin (CEA Saclay)

**Title**: 3D field theories with Chern-Simons term for large N in the Weyl gauge

**Abstract:** ADS/CFT correspondance has led to a number of conjectures concerning, conformal invariant, U(N) symmetric 3D field theories with Chern-Simons term for N large. An example is boson-fermion duality. This has prompted a number of calculations to shed extra light on the ADS/CFT correspondance.

We study here the example of gauge invariant fermion matter coupled to a Chern-Simons term. In contrast with previous calculations, which employ the light-cone gauge, we use the more conventional temporal gauge. We calculate several gauge invariant correlation functions. We consider general massive matter and determine the conditions for conformal invariance. We compare massless results with previous calculations, providing a check of gauge independence.

We examine also the possibility of spontaneous breaking of scale invariance and show that this requires the addition of an auxiliary scalar field.

Our method is based on field integral and steepest descent. The saddle point equations involve non-local fields and take the form of a set of integral equations that we solve exactly.

## 7. Other

Nothing to report.