Math-Phys Seminar
17 events
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Seminar
The index of lattice Dirac operators and K-theory
May 15 (Thu) at 13:30 - 15:00, 2025
Hidenori Fukaya (Assistant Professor, Department of Physics, Osaka University)
We show that the Wilson Dirac operator in lattice gauge theory can be identified as a mathematical object in K-theory and that its associated spectral flow is equal to the index. In comparison to the standard lattice Dirac operator index, our formulation does not require the Ginsparg-Wilson relation and has broader applicability to systems with boundaries and to the mod-two version of the indices in general dimensions. We numerically verify that the K and KO group formulas reproduce the known index theorems in continuum theory. We examine the Atiyah-Singer index on a flat two-dimensional torus and, for the first time, demonstrate that the Atiyah-Patodi-Singer index with nontrivial curved boundaries, as well as the mod-two versions, can be computed on a lattice (This seminar is co-organized with FQSP).
Venue: Seminar Room #359 (Main Venue) / via Zoom
Event Official Language: English
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Seminar
Foliation Matter Phase and Godbillon-Vey Invariant
October 16 (Wed) at 15:00 - 17:00, 2024
Taiichi Nakanishi (Ph.D. Student, Division of Physics and Astronomy, Graduate School of Science, Kyoto University)
It has been a main topic in today's physics to classify matter phases. Especially, topologically ordered phases are attracting much attension from broad perspective. However, most of mathematical structures other than the topology are not investigated yet in physics. In this talk, we present a physical model which is strongly connected to the foliation structure of the space manifold, and its field theoretical description. In such a foliation field theory, we can see the structure is highly connected to the mathematical invariant of foliation structures called Godbillon-Vey invariant. This work would be a fiest step toward shining a light on mathematical structures used in physics. This work is based on arXiv:2408.05048 with Hiromi Ebisu, Masazumi Honda, and Soichiro Shimamori.
Venue: Seminar Room #359 (Main Venue) / via Zoom
Event Official Language: English
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Seminar
Prefactorization algebra and theta term
May 21 (Tue) at 16:00 - 17:30, 2024
Masashi Kawahira (Ph.D. Student, Yukawa Institute for Theoretical Physics, Kyoto University)
Quantum field theories (QFTs) describe a lot of physical phenomena in our world. And giving a mathematical definition of QFTs is a long-standing problem. There are several mathematical formulations: Wightman formulation, Osterwalder–Schrader formulation and Atiyah-Segal formulation. And each of them cover different aspects of QFTs. Recently, Costello and their collabolators formulate QFTs by using prefactorization algbras. This formulaion cover a lot of classes of QFTs: TQFTs, 2d CFTs and perturbative QFTs. And they reproduce various results such as asymptotic freedom in non-Abelian gauge theories. Prefactorization algbras can be given by Batalin–Vilkovisky quantization (BV quantization) of the Lagrangian. However the original BV quantizations are perturbative and they do not have non-perturbative effects like instantons. In this talk, we propose the way to include Abelian-instanton effects. In modern language, it is the same as ℤgauging.
Venue: Seminar Room #359 (Main Venue) / via Zoom
Event Official Language: English
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Seminar
A Selective Survey of Ideas, Tools and Results in Constructive QFT II
April 1 (Mon) at 13:00 - 14:30, 2024
Christy Koji Kelly (Special Postdoctoral Researcher, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS))
In this talk we continue to discuss the constructive programme in rigorous QFT. Our aim is discuss a few points on difficulties in relation to the definition of Euclidean path integrals and the general strategy of constructive field theory from a constructive perspective. This will motivate a discussion of the fundamental results in probability theory which are the main tools for the construction of Euclidean path integral measures. We might also discuss some points in relation to free fields, including in particular difficulties in the characterisation of massless fields.
Venue: via Zoom / Seminar Room #359
Event Official Language: English
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Seminar
A Selective Survey of Ideas, Tools and Results in Constructive Field Theory
February 26 (Mon) at 13:00 - 14:30, 2024
Christy Koji Kelly (Special Postdoctoral Researcher, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS))
In this talk we discuss some aspects of constructive field theory with an emphasis on analytical and probabilistic methods and results. In particular after an overview of some points in the history of constructive QFT we plan to discuss some early achievements in axiomatic QFT, some features of the theory of distributions and the basic structure of the Wightman reconstruction theorem. We also introduce the Osterwalder-Schrader axioms and overview the strategy for the construction of nontrivial measures describing path-integrals for interacting QFTs. Depending on time constraints we might also discuss probabilistic tools (weak convergence of measures, the Bochner-Minlos theorem etc), Gaussian measures, UV regularity of simple QFTs and the construction of (infinite volume) Euclidean P(phi)_2 measures. The plan is to discuss some of these topics in some detail after the end of the official seminar.
Venue: via Zoom / Seminar Room #359
Event Official Language: English
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Seminar
Application of Modular tensor category to Lattice gauge theory
December 29 (Fri) at 10:30 - 16:00, 2023
Tomoya Hayata (Assistant Professor, Faculty of Economics, Keio University)
Inspired by the recent development in quantum computers, much efforts have been devoted to exploring their potential applications in lattice gauge theories. However, in contrast to condensed matter systems, we face many challenges in applications of quantum computations to lattice gauge theories, where one of the major obstructions lies in implementation of gauge symmetries in quantum computations. In this seminar, I talk about a possible solution to the problem based on a unitary modular tensor category, expressing the Hamiltonian of lattice gauge theories in terms of the so called F moves, and implementing the F moves on quantum computers. References: TH, Y. Hidaka, JHEP 09 (2023) 126; JHEP 09 (2023) 123.
Venue: Seminar Room #359
Event Official Language: English
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Seminar
Vortex Reconnection in Classical and Quantum Fluids
November 10 (Fri) at 15:00 - 16:30, 2023
Yoshifumi Kimura (Professor, Graduate School of Mathematics, Nagoya University)
As a fundamental process in turbulence, vortex reconnection has been studied widely not only in classical fluids but also in quantum fluids. For the latter, the first real observation of vortex reconnection was rather recent, and since then active analyses have been continued. On the other hand, vortex reconnection in the former has a long history, and it is now studied intensively as a candidate for a solution to the problem of the regularity/Singularity of the Navier-Stokes equations, which is one of the seven millennium prize problems of the Clay Mathematical Institute. In this talk, after introducing the problem, we will present some results of analysis and simulations of the dynamical system which has been proposed to describe a vortex reconnection of two vortex rings located symmetrically on two tilted planes [1], [2], [3]. Then it will be shown that this dynamical system can be written in noncanonical Hamiltonian form with Hamiltonian, H, and a Casimir invariant, C in the limit of zero viscosity [4].
Venue: Seminar Room #359 (Main Venue) / via Zoom
Event Official Language: English
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Seminar
Feynman’s proof of integrability of Calogero system from a modern point of view
March 10 (Fri) at 10:00 - 11:30, 2023
Yehao Zhou (Project Researcher, Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), The University of Tokyo)
In his last year of life Feynman was interested in integrable system, and in his study of Calogero models he came up with his own proof of the commutativity of integrals of motions of these models, which remains unpublished until it was transcribed by Polychronakos in 2018. His idea is to organize integrals of motions of a Calogero model into a generating function of differential operators which look like a correlation function in a certain free theory, then he showed that the generating function of differential operators commute for all spectral values, which leads to a proof of commutativity of integrals of motions. He commented on his proof “I learn nothing, no real clue as to why all this works, and what it means”. Recently in a joint work with Davide Gaiotto and Miroslav Rapcek we identify Feynman’s generating function as the correlation function of Miura operators in a W-algebra of type A, and in the rational and trigonometric cases we show that they equal to certain elements in the Dunkl representation of corresponding spherical Cherednik algebras in type A, which make the commutativity self-evident. This progress is a byproduct of a project in the study of M2-M5 brane junction in the M-theory.
Venue: Common Room #246-248 / via Zoom
Event Official Language: English
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Seminar
String theory, N=4 SYM and Riemann hypothesis
February 16 (Thu) at 14:00 - 16:10, 2023
Masazumi Honda (Assistant Professor, Yukawa Institute for Theoretical Physics, Kyoto University)
We discuss new relations among string theory, four-dimensional N=4 supersymmetric Yang-Mills theory (SYM) and the Riemann hypothesis. It is known that the Riemann hypothesis is equivalent to an inequality for the sum of divisors function σ(n). Based on previous results in literature, we focus on the fact that σ(n) appears in a problem of counting supersymmetric states in the N=4 SYM with SU(3) gauge group: the Schur limit of the superconformal index plays a role of a generating function of σ(n). Then assuming the Riemann hypothesis gives bounds on information on the 1/8-BPS states in the N=4 SYM. The AdS/CFT correspondence further connects the Riemann hypothesis to the type IIB superstring theory on AdS5×S5. In particular, the Riemann hypothesis implies a miraculous cancellation among Kaluza-Klein modes of the supergravity multiplet and D3-branes wrapping supersymmetric cycles in the string theory. We also discuss possibilities to gain new insights on the Riemann hypothesis from the physics side. This talk is based on a collaboration with Takuya Yoda (arXiv:220317091).
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
An Introduction to Rough Geometry (with a view to Euclidean Gravity)
October 14 (Fri) at 14:00 - 16:30, 2022
Christy Koji Kelly (Special Postdoctoral Researcher, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS))
The mathematical formulation of Einstein gravity typically utilises differentiable manifolds as models of smooth spacetimes. In many scenarios, however, it is desirable to have coarser models of spacetime and a correspondingly rough theory of geometry applicable to these coarser spacetime structures. In 2D Euclidean quantum gravity, for instance, the use of Regge calculus allows one to treat triangulations as regularisations of smooth spacetimes. There has been much recent progress in the mathematical (rigorous) understanding of this theory which we briefly review. We also introduce a rich alternative framework for the study coarse Euclidean geometry in the form of metric geometry augmented by optimal transport theory. In particular we introduce several optimal transport theoretic curvatures and demonstrate that these recover the familiar smooth notions under suitable limits.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Implications of singularity theorem for non-singular universe
June 16 (Thu) at 13:30 - 15:00, 2022
Daisuke Yoshida (Designated Assistant Professor, Graduate School of Mathematics, Nagoya University)
The singularity theorem by Penrose shows that a spacetime singularity arises in certain universal situations. The existence of a spacetime singularity is thought to represent a breakdown in the validity of theories such as general relativity and the phenomenological models of the universe. Thus, if we could build a correct model that describes the beginning of the universe, the universe predicted by that model should be non-singular. In this talk, we will discuss general properties that a non-singular universe must satisfy in order to avoid the singularity theorem. In particular, we will see that the universe must be, in some sense, smaller than the corresponding closed de Sitter spacetime.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Recent Progress in the Swampland Program
May 19 (Thu) at 14:00 - 15:30, 2022
Toshifumi Noumi (Associate Professor, Institute of Cosmophysics, Department of Physics, Graduate School of Science, Kobe University)
In the past years, it has become increasingly clear that there exist nontrivial consistency conditions on symmetries in quantum gravity, that are invisible in classical gravity. The Swampland program aims at identifying such quantum gravity constraints and their implications for particle physics and cosmology, toward quantum gravity phenomenology. In this talk, I will review recent progress in this program, including my own works.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Self-adjoint extension in quantum mechanics and non-Rydberg spectra of one-dimensional hydrogen atom
April 13 (Tue) at 16:00 - 18:10, 2021
Takuju Zen (Professor, School of Environmental Science and Engineering, Kochi University of Technology)
We offer a beginner’s guide to the functional-analytical techniques in quantum mechanics, and cover its application to the 1D Coulomb problem. It is shown that the wave function at the diverging point of the Coulomb potential is mathematically described by three-parameter family of generalized connection conditions. A scheme is devised to physically implement the generalized conditions, which provides the way to experimentally realize non-Rydberg spectra in 1D Hydrogen atom. Schedule: Part 1, Self-adjoint extension of Hilbert space operator Part 2, 1D Coulomb problem
Venue: via Zoom
Event Official Language: English
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Seminar
Non-perturbative tests of duality cascades in three dimensional supersymmetric gauge theories
December 14 (Mon) at 16:00 - 18:10, 2020
Naotaka Kubo (Postdoctoral Researcher, Yukawa Institute for Theoretical Physics, Kyoto University)
M2-brane is an interesting object in M-theory and string theory. A three-dimensional ?=6 super conformal Chern Simons theory with gauge group U(?1)×?(?2), called ABJ theory, describes the low energy behavior of M2-brane On the one hand, it has been considered that when |?1−?2| is larger than the absolute value of Chern Simons level, the supersymmetry is broken. On the other hand, it was predicted that an interesting phenomenon called duality cascade occurs, and supersymmetry is not broken in some cases. Motivated by this situation, we performed non-perturbative tests by focusing on the partitionfunction on ?3. The result strongly suggests that the duality cascade indeed occurs. We also proposed that the duality cascade occurs in theories with more general gauge groups and we performed non-perturbative tests in the same way. I will review and explain our physical prediction in the first half of my talk. In the second half of my talk , I will explain the non-perturbative tests . This part is mathematical because the partition function reduces to a matrix model by using the supersymmetric localization technique.
Venue: via Zoom
Event Official Language: English
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Seminar
Mathematics of thermalization in isolated quantum systems
November 10 (Tue) at 16:00 - 18:10, 2020
Naoto Shiraishi (Assistant Professor, Department of Physics, Faculty of Science, Gakushuin University)
If an isolated macroscopic quantum system is left at a nonequilibrium state, then this system will relax to the unique equilibrium state, which is called thermalization. Most of quantum many-body systems thermalize, while some many-body systems including integrable systems do not thermalize. What determines the presence/absence of thermalization and how to understand thermalization from microscopic quantum mechanics are profound long-standing problems. In the first part of my talk, I briefly review some established results of quantum thermalization. I first clarify the problem of thermalization in a mathematical manner, and then introduce several important results and insights: typicality of equilibrium states [1], relaxation caused by large effective dimension [2], and eigenstate thermalization hypothesis (ETH) [3,4] and weak-ETH [5]. In the second part of my talk, I explain some of my results. First, I introduce a model which is non-integrable and thermalizes but does not satisfy the ETH [6,7]. This finding disproves the conjectures that all nonintegrable systems satisfy the ETH and that the ETH is a necessary condition for thermalization. I also discuss the hardness of the problem of thermalization from the viewpoint of computational science [8]. Then, I move to an analytical approach to a concrete model, and prove that S=1/2 XYZ chain with a magnetic field is nonintegrable [9]. This is the first example of proof of nonintegrability in a concrete quantum many-body system, which will help a mathematical approach to thermalization.
Venue: via Zoom
Event Official Language: English
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Seminar
TQFT, integrable lattice model, and quiver gauge theories
October 2 (Fri) at 16:00 - 18:00, 2020
Toshihiro Ota (Student Trainee, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) / Ph.D. Student, Graduate School of Science, Osaka University)
1st part (math): In physics literature, “lattice models” appear quite often as mathematical models of physical systems, e.g. Ising model, vertex models, lattice gauge theory. The aim of the 1st part is to introduce ‘what is (T)QFT,’ ‘what is lattice model,’ and ‘what does integrability mean’ in the language of mathematics. In turn, they will play a crucial role in the 2nd part of my talk. I also hope that this will lead to a good exchange among us, especially between physicists and mathematicians. 2nd part (physics): In the 2nd part, I would like to explain where an integrable lattice model may come from, especially for people in the physics background. I will show a certain class of integrable lattice models is realized by Wilson-’t Hooft lines in 4d quiver gauge theories. I will also explain a bit how these gauge theories are constructed from brane configurations in string theory. String dualities allow us to relate the original 4d setups to 4d partially topological Chern-Simons theory, which is a partial TQFT and generates integrable lattice models. Please contact Keita Mikami's mail address to get access to the Zoom meeting room.
Venue: via Zoom
Event Official Language: English
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Seminar
Geometric Perspective for the Theory of Hydrodynamic Limits
August 31 (Mon) - September 1 (Tue), 2020
Makiko Sasada
Kenichi Bannai (Professor, Faculty of Science and Technology Department of Mathematics, Keio University)This is a series of lectures on "Geometric Perspectives for Fluid Dynamic Limit Theory" by the following speakers: [DAY 1: Aug 31] Dr. Makiko Sasada (University of Tokyo) [DAY 2: Sept 1] Prof. Kenichi Bannai (Keio University) Abstract: One of the fundamental problems in the natural and social sciences is to explain macroscopic phenomena that we can observe from the rules governing the microscopic system giving rise to the phenomena. Hydrodynamic limit provides a rigorous mathematical method to derive the deterministic partial differential equations describing the time evolution of macroscopic parameters, from the stochastic dynamics of a microscopic large scale interacting system. In the article "Topological Structures of Large Scale Interacting Systems via Uniform Locality" joint with Yukio Kametani, we introduce a general framework encompassing a wide variety of interacting systems in order to systematically investigate various microscopic stochastic large scale interacting systems in a unified fashion. In particular, we introduced a new cohomology theory called the uniformly local cohomology to investigate the underlying geometry of the interacting system. Our theory gives a new interpretation of the macroscopic parameters, the role played by the group action on the microscopic system, and the origin of the diffusion matrix associated to the macroscopic deterministic partial differential equation obtained via the space-time scaling limit of the microscopic system. The purpose of the series of lectures is to introduce to the audience the theory of hydrodynamic limits, especially the relation between the macroscopic observables and the microscopic interacting system. We then explain our new perspective of how geometry comes into play in investigating the interacting system, and introduce the ideas and results of our article. *Detailed information about the seminar refer to the email.
Venue: via Zoom
Event Official Language: English
17 events