FY2022 Annual Report
Gravity, Quantum Geometry and Field Theory Unit
Assistant Professor Reiko Toriumi
Abstract
Gravity, Quantum Geometry and Field Theory Unit studied the topics in quantum gravity. In particular, members are interested in random geometrical and field theoretical approaches both perturbative and nonperturbative methods. The research of the Unit intersects with a wide range of mathematics, including probability, combinatorics, topology, knot theory, reprentation theory, etc.
1. Staff
 Dr. Reiko Toriumi, Group Leader
 Dr. Nicolas Delporte, Postdoc
 Dr. Remi Cocou Avohou, Postdoc
 Dr. Rudrajit Banerjee, Postdoc
 Dr. Cihan Pazarbaşı, Postdoc
 Saswato Sen, Graduate Student
 Andreani Petrou, Graduate Student
 Juan Abranches, Graduate Student
 Matthias Vancraeynest, Research Intern
 Chikako Sugiyama, Administrative Assistant
 Yukiko Nakagawa, Administrative Assistant
 Chiyo Eto, Administrative Assistant
2. Collaborations
2.1 Scaling solutions for asymptotically safe gravity
 Description: Published.
 Type of collaboration: Joint research
 Researchers:
 Prof. Christof Wetterich, Heidelberg Univerity, Germany
 Dr. Masatoshi Yamada, Heidelberg University, Germany
 Saswato Sen, OIST
2.2 Random Fields and Random Geometries
 Description: Ongoing.
 Type of collaboration: Joint research
 Researchers:
 Prof. Reiko Toriumi, OIST
 Dr. Nicolas Delporte, OIST
 Saswato Sen, OIST
2.3 On ribbon configurations
 Description: Ongoing.
 Type of collaboration: Joint research
 Researchers:
 Prof. Vyacheslav Futorny (USP, Brazil)
 Prof. Kostiantyn Iusenko (USP, Brazil)
 Dr. Remi Avohou, OIST
Classifying graphs of higher
2.4 Classifying graphs for higher grades for multiorientable multimatrix modelClassifying graphs of higher
Classifying graphs of higher
 Description: Ongoing.
 Type of collaboration: Joint research
 Researchers:
 Prof. Reiko Toriumi, OIST
 Dr. Remi Avohou, OIST
 Matthias Vancraeynest, OIST
2.5 Gross, Mansour and Tucker conjecture for ∆matroidsGross, Mansour and Tucker conjecture for ∆matroidsGross, Mansour and Tucker conjecture for ∆matroidsClassifying graphs of higher
Classifying graphs of higher
 Description: Ongoing.
 Type of collaboration: Joint research
 Researchers:
 Prof. Fabien Vignes Tourneret, CNRS Lyon (France)
 Dr. Remi Avohou, OIST
ssifying graphs of higher
Classifying graphs of higher2.6 Computerization of the Recursive Derivative Expansion of Schrodinger/Heat KernelsClassifying graphs of higher
Classifying graphs of higher
 Description: Ongoing.
 Type of collaboration: Solo research
 Researchers:
 Dr. Cihan Pazarbaşı, OIST
2.7 Nonperturbative Aspects of Worldline QED
Classifying graphs of higher
 Description: Ongoing.
 Type of collaboration: Solo research
 Researchers:
 Dr. Cihan Pazarbaşı, OIST
2.8 Nonperturbative Aspects of Random Matrix Models
Classifying graphs of higher
 Description: Ongoing.
 Type of collaboration: Joınt research
 Researchers:
 Prof. Reiko Toriumi, OIST
 Dr. Nicolas Delporte, OIST
 Dr. Cihan Pazarbaşı, OIST

Julian De Vuyst, OIST
2.9 On the Solution to the CDT Matrix Model's Partition Function
Classifying graphs of higher
 Description: Ongoing.
 Type of collaboration: Joınt research
 Researchers:
 Prof. Reiko Toriumi, OIST
 Juan Abranches, OIST
2.10 JT gravity at finite cutoff
 Description: Ongoing.
 Type of collaboration: Joınt research
 Researchers:
 Prof. Frank Ferrari, ULB
 Dr. Romain Pascalie, ULB
 Dr. Nicolas Delporte, OIST
2.11 Eigenvalues of random tensors
 Description: Ongoing.
 Type of collaboration: Joınt research
 Researchers:
 Prof. Reiko Toriumi, OIST
 Prof. Benoit Collins, Kyoto U
 Prof. Naoki Sasakura, YITP
 Dr. Luca Lionni, Heidelberg University
 Dr. Nicolas Delporte, OIST
2.12 Ncutoff regularization for fields on hyperbolic space
 Description: Ongoing.
 Type of collaboration: Joınt research
 Researchers:
 Dr. Rudrajit Banerjee, OIST
 Dr. Maximilian Becker, Radboud University
 Renata Ferrero, JGU Mainz
2.13 States of Low Energy on cosmological spacetimes
 Description: Ongoing.
 Type of collaboration: Joınt research
 Researchers:
 Prof. Max Niedermaier, University of Pittsburgh
 Dr. Rudrajit Banerjee, OIST
2.14 Knot invariants and matrix models
 Description: Ongoing.
 Type of collaboration: Joınt research
 Researchers:
 Prof. Shinobu Hikami, OIST
 Prof. Reiko Toriumi, OIST
 Andreani Petrou, OIST
2.15 Oneloop betafunctions of quartic enhanced tensor field theories
 Description: Published.
 Type of collaboration: Joınt research
 Researchers:
 Prof. Reiko Toriumi, OIST
 Prof. Joseph Ben Geloun, University Paris 13
3. Activities and Findings
3.1 Scaling solutions for asymptotically safe gravity
The project was initiated by Saswato Sen as a master student in Heidelberg University with Dr. Yamada and continued after joining OIST as a research intern and then as a rotation student. Within a general truncation of the functional flow equations for quantum gravity with up to four derivatives of the metric we have demonstrated the existence of a scaling solution or critical trajectory from the asymptotically free ultraviolet fixed point to the infrared fixed point.
3.2 Random fields and Random geometries
We are developing an approach to understand quantum matter on random graphs. We use correspondence of random walk statistics and matter correlators to understand scattering amplitudes of field theories on graphs. We have identified a suitable random walk which corresponds to fermionic fields on graphs. We are studying the behaviour of such a random walker on various geometries.
3.3 On ribbon configurations
We review the Brauer configuration algebra. Some results about nangulation algebra, which generalizes the triangulation algebra, are obtained. We also found a full formula for the Cartan matrix of the Brauer algebra, allowing us to characterize the Brauer configuration of affine and finite type.
3.4 Classifying Graphs for higher grades for multiorientable multimatrix model
The key objective of this work is to extend some recent findings from [Ann.Inst.H.Poincare D Comb.Phys.Interact., 2022, 9 (2), pp.367433]. This paper describes how to build all of the grade $l=0$ melonfree Feynman graphs of genus $g$ from the family of grade $l=0$ melonfree Feynman graphs of genus $g'<g$. This study finds a reasonable generalization for grade $l=1$ melonfree Feynman graphs of genus $g$, which we hope to generalize to arbitrary grade melonfree Feynman graphs.
3.5 Gross, Mansour and Tucker conjecture for ∆matroids
The classical EulerPoincaré duality is generalized by the partial duality of ribbon graphs with respect to a subset of their edges. This operation frequently alters the genus. J.L. Gross, T. Mansour, and T.W. Tucker recently proposed that for any ribbon graph other than plane trees and their partial duals, there is a subset of edges whose partial duality changes the genus. Qi Yan and Xian'an Jin discovered a family of counterexamples. S. Chmutov and F. VignesTourneret show that these are the only counterexamples, and they wonder whether the partialdual genus polynomial and related conjectures would make sense for deltamatroids. The primary purpose of this research is to provide an answer to such a query.
3.6 Computerization of the Recursive Derivative Expansion of Schrodinger/Heat Kernels
Heat Kernels or their Wick rotated counterpart Schrodinger kernels (time evolution propagators) are two of the central objects in quantum mechanics and worldline QFT problems. In this project, in order to compute high order perturbative expansions of these objects, we constructed an efficient algorithm based on a recursive derivative expansion, which is an organized version of the Schwinger DeWitt expansion. Currently, we are working on its generalization for practical uses in application to various problems in quantum mechanics, abalian gauge theories and curved space time QFT.
3.7 Nonperturbative Aspects of Worldline QED
Based on the worldline formalism, we investigated the nonperturbative properties of the QED vacuum. Using high order perturbative expansions and their BorelPade summation, we obtained nonperturbative contributions to "beyond" the leading order for arbitrary background potentials. We also properly established a previously unnoticed relationship between the sources of nonperturbative information of the worldline QED and the ones obtained from (exact) WKB method. We plan to use this observation and extend our investigation to various projects on the nonperturbative and resurgence properties of various problems on quantum mechanics and worldline QFT.
3.8 Nonperturbative Aspects of Random Matrix Models
We reviewed the nonperturbative aspects of Hermitian matrix models at 1 cut and 2 cuts using their eigenvalue decomposition. We also worked on the relationship between the statistical physics of the eigenvalues and effective quantum mechanics which governs their dynamics. We are currently investigating the nonperturbative aspects of the effective quantum mechanics in relation with the observables of the random matrix models in various limits including the doublescaling limit where the observables are associated to 2D quantum gravity.
3.9 On the Solution to the CDT Matrix Model's Partition Function
We explore different solution methods for hermitian matrix models without unitary symmetry, with the goal of applying it to the unsolved problem of the partition function of the CDT matrix model. We work with representation theory properties such as character expansion and SchurWeyl duality. We are able to find a solution in the large N limit, with the possible extension to finite N. This project started as an internship project that was continued as a rotation project, and is currently on writing process.
3.10 JT gravity at finite cutoff
We use analytical and numerical tools to characterise the phase space of twodimensional geometries with boundary and fixed curvature, in terms of properties of their boundaries (corresponding to selfoverlapping curves). We aim at making precise the regimes where the Schwarzian approximation (for hyperbolic metrics) and selfavoiding walk hold and what lies beyond. These curves are still relatively new objects and good sets of parameters are being searched for.
3.11 Eigenvalues of random tensors
We try bridge different approaches to eigenvalues of random tensors and using field theoretic techniques (sypersymmetric formalism and largeN approximations) to derive the eigenvalue distribution at leading order, the law of the largest eigenvalue, their correlations. Subleading orders are also important to see how far any relation to free probability can go.
3.12 Ncutoff regularization for fields on hyperbolic space
We investigate a novel selfconsistent quantization scheme, the Ncutoffs, for scalar and metric fluctuations on the maximally symmetric but noncompact hyperbolic space. The Ncutoffs are a regularization on the spectrum of the fields' fluctuation modes, and we find that the inclusion of increasingly many modes tends to drive the the negative curvature of hyperbolic space towards zero, leading to vanishing values when the cutoff is removed.
3.13 States of Low Energy on cosmological spacetimes
States of Low Energy are exact Hadamard states for free quantum fields on FriedmannRobertsonWalker spacetimes. We extend this construction to a wide class of spacetimes relevant for primordial cosmology, where in addition to proving the Hadamard property, systematic series expansions in the infrared and ultraviolet regimes are developed.
3.14 Knot matrix models and skein theory
Knot matrix models are defined by equating knot polynomial invaianrts to averages of characters of representations. However, an explicit measure for the average is only known for the special case of torus knots and its generalization to other families of knots remains an open problem. Our aim is to extend this definition to hyperbolic knots, by exploiting mirror dymmetry between unitary and Hermitian ensembles. In parallel, we are trying to construct a recursive formula for the HOMFLYPT polynomial of torus knots via skein theory, which may be given an interpretation as creation/annhilation of fermions and bosons.
3.15 Oneloop betafunctions of quartic enhanced tensor field theories
Enhanced tensor field theories (eTFT) have dominant graphs that do not correspond to melonic diagrams of ordinary tensor field theories. They therefore describe pertinent candidates to escape the socalled branched polymer phase, the universal geometry found for tensor models. For generic rank d of the tensor field, we compute the perturbative betafunctions at oneloop of two justrenormalizable quartic eTFT coined by + or x, depending on their vertex weights. The models + has two quartic coupling constants $(\lambda, \lambda_+)$, and two 2point couplings (mass, $Z_a$). Meanwhile, the model x has two quartic coupling constants $(\lambda, \lambda_x)$ and three 2point couplings (mass, $Z_a$, $Z_{2a}$). At all orders, both models have a constant wave function renormalization: $Z=1$ and therefore no anomalous dimension. Despite such peculiar behavior, both models acquire nontrivial radiative corrections for the coupling constants. The RG flow of the model $+$ exhibits neither asymptotic freedom nor the ordinary Landau ghost of phi^4_4 model: \lambda_+ is a fixed point and \lambda has linear behavior in time scale $t = \log(k/k_0)$. In the UV, the mass behaves likewise namely is linear in t, whereas $Z_a$ decreases exponentially towards a constant value. For the model x, both $\lambda$ and $\lambda_x$ do not flow, all remaining 2point coupling constants are linear functions of the time scale in the UV.
4. Publications
4.1 Journals
 Sen, S.;Wetterich, C.; Yamada, M. Scaling solutions for asymptotically safe gravity JHEP 02 (2023) 054 . Scaling solutions for asymptotically safe gravity, DOI: 10.1007/JHEP02(2023)054
 Ben Geloun, J.;Toriumi, R. Oneloop betafunctions of quartic enhanced tensor field theories arXiv:2303.09829 DOI:10.48550/arXiv.2303.09829
 Martini, R.;Toriumi, R. Trisections in colored tensor models, accepted to Annales de l'Institut Henri Poincare D: Combinatorics, Physics and their interactions DOI:10.4171/AIHPD/167
4.2 Books and other onetime publications
Nothing to report
4.3 Oral and Poster Presentations
([NOTE] *Seminars and workshops by OIST faculty/unit members (either with or without other speakers), either at OIST or at other institutions than OIST, should be included in the 4.3 Oral and Poster Presentations.
 Remi C. Avohou, From Brauer graph algebra to Brauer configuration algebra, workshop "Women at the intersection of mathematics and theoretical physics meet in Okinawa", Okinawa, Japan, 2024 March 2023.
 Andreani Petrou, Towards knot matrix models for families of twisted hyperbolic knots, Silver Workshop V, Okinawa, Japan, Country, 911 November 2022.
 Andreani Petrou, Towards knot matrix models for families of twisted hyperbolic knots, workshop "Women at the intersection of mathematics and theoretical physics meet in Okinawa", Okinawa, Japan, 2024 March 2023.
 Nicolas Delporte, A random walk approach to two dimensional quantum gravity, invited seminar, LIPN, Paris, France, 14 February 2023.
 Nicolas Delporte, On aspects of twodimensional quantum gravity, invited seminar, LIPN, Paris, France, 14 February 2023.
 Nicolas Delporte, Peeking at quantum gravity with selfoverlapping curves, invited seminar, IHP, Paris, France, 15 February 2023.
 Reiko Toriumi, Trisections in colored tensor models, workshop "Random Geometry in Heidelberg", Heidelberg University, Germany 1620 May 2022.
 Reiko Toriumi, Trisections in colored tensor models, workshop "Quantum Gravity and Random Geometry", Institute Henri Poincare, Paris, France 1620 January 2023.
 Rudrajit Banerjee, The spatial Functional Renormalization Group and Hadamard states on cosmological spacetimes, conference "Pheno 2022", University of Pittsburgh, USA, 911 May 2022.
 Rudrajit Banerjee, Wick rotating the heat kernel, invited seminar, IMAPP Radboud University, Nijmegen, The Netherlands, 24th February 2023.
 Rudrajit Banerjee, Wick rotating the heat kernel, invited seminar, JGU Mainz, Germany, 28th February 2023.
5. Intellectual Property Rights and Other Specific Achievements
Nothing to report
6. Meetings and Events
6.1 Vortex counting, WKB and the higher order Stokes phenomenon
 Date: March 28, 2023
 Venue: OIST Campus Lab4 E01
 Speaker: Dr. Samuel Crew (University of Bath)
6.2 An introduction to combinatorial compact quantum groups and free Bessel distributions
 Date: March 27, 2023
 Venue: OIST Campus Lab4 F01
 Speaker: Prof. Benoit Collins (Kyoto University)
6.3 Women at the Intersection of Mathematics and Theoretical Physics Meet in Okinawa
 Date: March 20 to 24, 2023
 Venue: OIST Campus Lab4 E48
 Coorganizers:
 Dr. Shihoko Ishii (University of Tokyo)
 Dr. Sylvie Paycha (University of Potsdam)
 Dr. Susanne Reffert (University of Bern)
 Dr. Kasia Rejzner (University of York)
 Dr. Xiaodan Zhou (OIST)
 Speakers:
 Dr. Yuanyuan Bao (University of Tokyo, Japan)
 Dr. Xenia de la Ossa (Oxford University, UK)
 Dr. Yukari Ito (IPMU, Japan)
 Dr. Motoko Kato (Ryukyu University, Japan)
 Dr. Keiko Kawamuro (University of Iowa, USA)
 Dr. Yukiko Konishi (Tsuda Univesity, Japan)
 Dr. Yolanda Lozano (Univesity of Oviedo, Spain)
 Dr. Chihiro Matsui (University of Tokyo, Japan)
 Dr. Silvia Penati (INFN MilanoBiccoca, Italy)
 Dr. Makiko Sasada (University of Tokyo, Japan)
 Dr. Simona Settepanella (University of Torino, Italy)
 Dr. Mayuko Yamashita (RIMS, Kyoto, Japan)
6.4 Refined BPS invariants via topological recursion
 Date: November 25, 2022
 Venue: OIST Campus Lab4 F01
 Speaker: Dr. Kento Osuga (University of Sheffield, UK)
6.5 Invitation to topological recursion and their applications
 Date: November 17, 2022
 Venue: OIST Campus Lab4 F01
 Speaker: Dr. Kento Osuga (University of Sheffield, UK)
6.6 LandauGinzburg analysis of (T)GFT models
 Date: September 15, 2022
 Venue: OIST Campus Lab4 E01
 Speaker: Dr. Luca Marchetti (LudwigMaximiliansUniversität München)
6.7 Parametric resurgent transseries and large N vortex counting
 Date: August 22, 2022
 Venue: OIST Campus Lab4 F01
 Speaker: Dr. Samuel Crew (University of Bath)
6.8 Combinatorics and knot invariants
 Date: June 24, 2022
 Venue: OIST Campus Lab4 F01
 Speaker: Dr. Robert Osburn (University College Dublin, Ireland)
7. Other
Women in Mathematics Photography Exhibits
 Date: March 20 to 24, 2023
 Venue: OIST Central Building LevelC, Tunnel Gallery
 Photographer: Noel Tovia Matoff