Wall-crossing Structures in Cluster Algebras PDF Download
Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download Wall-crossing Structures in Cluster Algebras PDF full book. Access full book title Wall-crossing Structures in Cluster Algebras by Lang Mou. Download full books in PDF and EPUB format.
Author: Lang Mou
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Get Book Here
Book Description
In this dissertation, we study the phenomenon of wall-crossing structures in cluster algebras of Fomin and Zelevinsky, with examples including cluster scattering diagrams of Gross, Hacking, Keel, and Kontsevich (GHKK) and stability scattering diagrams of Bridgeland. We show that in general, every consistent scattering diagram admits a canonical underlying cone complex structure. We describe mutations of the stability scattering diagram of a quiver with non-degenerate potential. Then we use this description to prove that the stability scattering diagram admits the so-called cluster complex structure. As a consequence, we verify if a quiver admits a reddening sequence, a conjecture of Kontsevich and Soibelman that the associated cluster scattering diagram is equivalent to the stability scattering diagram of the same quiver with a non-degenerate potential. We also give another proof of the Caldero-Chapoton formula of cluster monomials using scattering diagrams. Skew-symmetrizable cluster algebras need extra care. We define a Langlands dual version of the cluster scattering diagram of GHKK and show that it admits a cluster complex structure that is Langlands dual to GHKK's version. We use it to describe the cluster monomials of skew- symmetrizable cluster algebras in terms of theta functions. Then we study the Hall algebra scattering diagram associated to the Geiss-Leclerc-Schröer algebra of an acyclic skew-symmetrizable matrix with a skew-symmetrizer. We show that it admits the same cluster complex structure as the aforementioned Langlands dual cluster scattering diagram. In the end, we extend the theory of scattering diagrams to Chekhov and Shapiro's generalized cluster algebras.
Author: Lang Mou
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Get Book Here
Book Description
In this dissertation, we study the phenomenon of wall-crossing structures in cluster algebras of Fomin and Zelevinsky, with examples including cluster scattering diagrams of Gross, Hacking, Keel, and Kontsevich (GHKK) and stability scattering diagrams of Bridgeland. We show that in general, every consistent scattering diagram admits a canonical underlying cone complex structure. We describe mutations of the stability scattering diagram of a quiver with non-degenerate potential. Then we use this description to prove that the stability scattering diagram admits the so-called cluster complex structure. As a consequence, we verify if a quiver admits a reddening sequence, a conjecture of Kontsevich and Soibelman that the associated cluster scattering diagram is equivalent to the stability scattering diagram of the same quiver with a non-degenerate potential. We also give another proof of the Caldero-Chapoton formula of cluster monomials using scattering diagrams. Skew-symmetrizable cluster algebras need extra care. We define a Langlands dual version of the cluster scattering diagram of GHKK and show that it admits a cluster complex structure that is Langlands dual to GHKK's version. We use it to describe the cluster monomials of skew- symmetrizable cluster algebras in terms of theta functions. Then we study the Hall algebra scattering diagram associated to the Geiss-Leclerc-Schröer algebra of an acyclic skew-symmetrizable matrix with a skew-symmetrizer. We show that it admits the same cluster complex structure as the aforementioned Langlands dual cluster scattering diagram. In the end, we extend the theory of scattering diagrams to Chekhov and Shapiro's generalized cluster algebras.
Author: Ricardo Castano-Bernard
Publisher: Springer
ISBN: 3319065149
Category : Mathematics
Languages : en
Pages : 445
Get Book Here
Book Description
The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as “degenerations” of the corresponding algebro-geometric objects.
Author: Matteo Parisi
Publisher: Springer Nature
ISBN: 3031410696
Category : Science
Languages : en
Pages : 237
Get Book Here
Book Description
This book is a significant contribution within and across High Energy Physics and Algebraic Combinatorics. It is at the forefront of the recent paradigm shift according to which physical observables emerge from geometry and combinatorics. It is the first book on the amplituhedron, which encodes the scattering amplitudes of N=4 Yang-Mills theory, a cousin of the theory of strong interactions of quarks and gluons. Amplituhedra are generalizations of polytopes inside the Grassmannian, and they build on the theory of total positivity and oriented matroids. This book unveils many new combinatorial structures of the amplituhedron and introduces a new important related object, the momentum amplituhedron. Moreover, the work pioneers the connection between amplituhedra, cluster algebras and tropical geometry. Combining extensive introductions with proofs and examples, it is a valuable resource for researchers investigating geometrical structures emerging from physics for some time to come.
Author: Radu Laza
Publisher: Springer
ISBN: 1493928309
Category : Mathematics
Languages : en
Pages : 542
Get Book Here
Book Description
This volume presents a lively introduction to the rapidly developing and vast research areas surrounding Calabi–Yau varieties and string theory. With its coverage of the various perspectives of a wide area of topics such as Hodge theory, Gross–Siebert program, moduli problems, toric approach, and arithmetic aspects, the book gives a comprehensive overview of the current streams of mathematical research in the area. The contributions in this book are based on lectures that took place during workshops with the following thematic titles: “Modular Forms Around String Theory,” “Enumerative Geometry and Calabi–Yau Varieties,” “Physics Around Mirror Symmetry,” “Hodge Theory in String Theory.” The book is ideal for graduate students and researchers learning about Calabi–Yau varieties as well as physics students and string theorists who wish to learn the mathematics behind these varieties.
Author: Jörg Teschner
Publisher: Springer
ISBN: 3319187694
Category : Science
Languages : en
Pages : 467
Get Book Here
Book Description
This book reviews a number of spectacular advances that have been made in the study of supersymmetric quantum field theories in the last few years. Highlights include exact calculations of Wilson loop expectation values, and highly nontrivial quantitative checks of the long-standing electric-magnetic duality conjectures The book starts with an introductory article presenting a survey of recent advances, aimed at a wide audience with a background and interest in theoretical physics. The following articles are written for advanced students and researchers in quantum field theory, string theory and mathematical physics, our goal being to familiarize these readers with the forefront of current research. The topics covered include recent advances in the classification and vacuum structure of large families of N=2 supersymmetric field theories, followed by an extensive discussion of the localisation method, one of the most powerful tools for exact studies of supersymmetric field theories. The quantities that have been studied in this way are partition functions, expectation values of line operators, and supersymmetric indices. The book also reviews recently discovered connections between SUSY field theories in four dimensions and two-dimensional conformal field theory. These connections have a counterpart in relations between three-dimensional gauge theories and Chern-Simons theory; the book’s closing chapters explore connections with string theory.
Author: Ron Donagi
Publisher: American Mathematical Society
ISBN: 1470472406
Category : Mathematics
Languages : en
Pages : 306
Get Book Here
Book Description
This is a proceedings volume from the String-Math conference which took place at the University of Warsaw in 2022. This 12th String-Math conference focused on several research areas actively developing these days. They included generalized (categorical) symmetries in quantum field theory and their relation to topological phases of matter; formal aspects of quantum field theory, in particular twisted holography; various developments in supersymmetric gauge theories, BPS counting and Donaldson–Thomas invariants. Other topics discussed at this conference included new advances in Gromov–Witten theory, curve counting, and Calabi–Yau manifolds. Another broad topic concerned algebraic aspects of conformal field theory, vertex operator algebras, and quantum groups. Furthermore, several other recent developments were presented during the conference, such as understanding the role of operator algebras in the presence of gravity, derivation of gauge-string duality, complexity of black holes, or mathematical aspects of the amplituhedron. This proceedings volume contains articles summarizing 14 conference lectures, devoted to the above topics.
Author: Denis Auroux
Publisher: Birkhäuser
ISBN: 3319599399
Category : Mathematics
Languages : en
Pages : 368
Get Book Here
Book Description
This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim’s vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim’s heart. Contributors: S. Donaldson A. Goncharov D. Kaledin M. Kapranov A. Kapustin L. Katzarkov A. Noll P. Pandit S. Pimenov J. Ren P. Seidel C. Simpson Y. Soibelman R. Thorngren
Author: Jan de Gier
Publisher: Springer Nature
ISBN: 3030624978
Category : Mathematics
Languages : en
Pages : 798
Get Book Here
Book Description
MATRIX is Australia’s international and residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-4 weeks in duration. This book is a scientific record of the ten programs held at MATRIX in 2019 and the two programs held in January 2020: · Topology of Manifolds: Interactions Between High and Low Dimensions · Australian-German Workshop on Differential Geometry in the Large · Aperiodic Order meets Number Theory · Ergodic Theory, Diophantine Approximation and Related Topics · Influencing Public Health Policy with Data-informed Mathematical Models of Infectious Diseases · International Workshop on Spatial Statistics · Mathematics of Physiological Rhythms · Conservation Laws, Interfaces and Mixing · Structural Graph Theory Downunder · Tropical Geometry and Mirror Symmetry · Early Career Researchers Workshop on Geometric Analysis and PDEs · Harmonic Analysis and Dispersive PDEs: Problems and Progress The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on a topic related to the MATRIX program; the remaining contributions are predominantly lecture notes or short articles based on talks or activities at MATRIX.
Author: Robert J. Marsh
Publisher: Erich Schmidt Verlag GmbH & Co. KG
ISBN: 9783037191309
Category : Cluster algebras
Languages : en
Pages : 132
Get Book Here
Book Description
Cluster algebras are combinatorially defined commutative algebras which were introduced by S. Fomin and A. Zelevinsky as a tool for studying the dual canonical basis of a quantized enveloping algebra and totally positive matrices. The aim of these notes is to give an introduction to cluster algebras which is accessible to graduate students or researchers interested in learning more about the field while giving a taste of the wide connections between cluster algebras and other areas of mathematics. The approach taken emphasizes combinatorial and geometric aspects of cluster algebras. Cluster algebras of finite type are classified by the Dynkin diagrams, so a short introduction to reflection groups is given in order to describe this and the corresponding generalized associahedra. A discussion of cluster algebra periodicity, which has a close relationship with discrete integrable systems, is included. This book ends with a description of the cluster algebras of finite mutation type and the cluster structure of the homogeneous coordinate ring of the Grassmannian, both of which have a beautiful description in terms of combinatorial geometry.
Author: Cheol-Hyun Cho
Publisher: American Mathematical Society
ISBN: 1470447614
Category : Mathematics
Languages : en
Pages : 116
Get Book Here
Book Description
View the abstract.
