Instanton Counting Quantum Geometry And Algebra

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Instanton Counting, Quantum Geometry and Algebra

Taro Kimura
Instanton Counting Quantum Geometry and Algebra Book PDF

Author : Taro Kimura
Publisher : Springer Nature
Page : 297 pages
File Size : 53,5 Mb
Release : 2021-07-05
Category : Science
ISBN : 9783030761905

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Instanton Counting, Quantum Geometry and Algebra by Taro Kimura Pdf

This book pedagogically describes recent developments in gauge theory, in particular four-dimensional N = 2 supersymmetric gauge theory, in relation to various fields in mathematics, including algebraic geometry, geometric representation theory, vertex operator algebras. The key concept is the instanton, which is a solution to the anti-self-dual Yang–Mills equation in four dimensions. In the first part of the book, starting with the systematic description of the instanton, how to integrate out the instanton moduli space is explained together with the equivariant localization formula. It is then illustrated that this formalism is generalized to various situations, including quiver and fractional quiver gauge theory, supergroup gauge theory. The second part of the book is devoted to the algebraic geometric description of supersymmetric gauge theory, known as the Seiberg–Witten theory, together with string/M-theory point of view. Based on its relation to integrable systems, how to quantize such a geometric structure via the Ω-deformation of gauge theory is addressed. The third part of the book focuses on the quantum algebraic structure of supersymmetric gauge theory. After introducing the free field realization of gauge theory, the underlying infinite dimensional algebraic structure is discussed with emphasis on the connection with representation theory of quiver, which leads to the notion of quiver W-algebra. It is then clarified that such a gauge theory construction of the algebra naturally gives rise to further affinization and elliptic deformation of W-algebra.

Lie Theory and Its Applications in Physics

Vladimir Dobrev
Lie Theory and Its Applications in Physics Book PDF

Author : Vladimir Dobrev
Publisher : Springer Nature
Page : 526 pages
File Size : 43,8 Mb
Release : 2023-01-29
Category : Mathematics
ISBN : 9789811947513

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Lie Theory and Its Applications in Physics by Vladimir Dobrev Pdf

This volume presents modern trends in the area of symmetries and their applications based on contributions to the Workshop "Lie Theory and Its Applications in Physics" held in Sofia, Bulgaria (on-line) in June 2021. Traditionally, Lie theory is a tool to build mathematical models for physical systems. Recently, the trend is towards geometrization of the mathematical description of physical systems and objects. A geometric approach to a system yields in general some notion of symmetry which is very helpful in understanding its structure. Geometrization and symmetries are meant in their widest sense, i.e., representation theory, algebraic geometry, number theory, infinite-dimensional Lie algebras and groups, superalgebras and supergroups, groups and quantum groups, noncommutative geometry, symmetries of linear and nonlinear partial differential operators, special functions, and others. Furthermore, the necessary tools from functional analysis are included. This is a big interdisciplinary and interrelated field. The topics covered in this Volume are the most modern trends in the field of the Workshop: Representation Theory, Symmetries in String Theories, Symmetries in Gravity Theories, Supergravity, Conformal Field Theory, Integrable Systems, Quantum Computing and Deep Learning, Entanglement, Applications to Quantum Theory, Exceptional quantum algebra for the standard model of particle physics, Gauge Theories and Applications, Structures on Lie Groups and Lie Algebras. This book is suitable for a broad audience of mathematicians, mathematical physicists, and theoretical physicists, including researchers and graduate students interested in Lie Theory.

The Theory and Applications of Instanton Calculations

Manu Paranjape
The Theory and Applications of Instanton Calculations Book PDF

Author : Manu Paranjape
Publisher : Cambridge University Press
Page : 325 pages
File Size : 48,8 Mb
Release : 2017-11-16
Category : Science
ISBN : 9781107155473

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The Theory and Applications of Instanton Calculations by Manu Paranjape Pdf

This book explains in detail the key concepts, calculations and applications elucidating quantum tunnelling mediated by instantons, using the Feynman path integral.

Elementary Introduction to Quantum Geometry

Jan Ambjorn
Elementary Introduction to Quantum Geometry Book PDF

Author : Jan Ambjorn
Publisher : CRC Press
Page : 292 pages
File Size : 52,7 Mb
Release : 2022-11-02
Category : Mathematics
ISBN : 9781000776003

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Elementary Introduction to Quantum Geometry by Jan Ambjorn Pdf

This graduate textbook provides an introduction to quantum gravity, when spacetime is two-dimensional. The quantization of gravity is the main missing piece of theoretical physics, but in two dimensions it can be done explicitly with elementary mathematical tools, but it still has most of the conceptional riddles present in higher dimensional (not yet known) quantum gravity. It provides an introduction to a very interdisciplinary field, uniting physics (quantum geometry) and mathematics (combinatorics) in a non-technical way, requiring no prior knowledge of quantum field theory or general relativity. Using the path integral, the chapters provide self-contained descriptions of random walks, random trees and random surfaces as statistical systems where the free relativistic particle, the relativistic bosonic string and two-dimensional quantum gravity are obtained as scaling limits at phase transition points of these statistical systems. The geometric nature of the theories allows one to perform the path integral by counting geometries. In this way the quantization of geometry becomes closely linked to the mathematical fields of combinatorics and probability theory. By counting the geometries, it is shown that the two-dimensional quantum world is fractal at all scales unless one imposes restrictions on the geometries. It is also discussed in simple terms how quantum geometry and quantum matter can interact strongly and change the properties both of the geometries and of the matter systems. It requires only basic undergraduate knowledge of classical mechanics, statistical mechanics and quantum mechanics, as well as some basic knowledge of mathematics at undergraduate level. It will be an ideal textbook for graduate students in theoretical and statistical physics and mathematics studying quantum gravity and quantum geometry. Key features: Presents the first elementary introduction to quantum geometry Explores how to understand quantum geometry without prior knowledge beyond bachelor level physics and mathematics. Contains exercises, problems and solutions to supplement and enhance learning

Topological Recursion and its Influence in Analysis, Geometry, and Topology

Chiu-Chu Melissa Liu,Motohico Mulase
Topological Recursion and its Influence in Analysis Geometry and Topology Book PDF

Author : Chiu-Chu Melissa Liu,Motohico Mulase
Publisher : American Mathematical Soc.
Page : 549 pages
File Size : 44,7 Mb
Release : 2018-11-19
Category : Topology
ISBN : 9781470435417

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Topological Recursion and its Influence in Analysis, Geometry, and Topology by Chiu-Chu Melissa Liu,Motohico Mulase Pdf

This volume contains the proceedings of the 2016 AMS von Neumann Symposium on Topological Recursion and its Influence in Analysis, Geometry, and Topology, which was held from July 4–8, 2016, at the Hilton Charlotte University Place, Charlotte, North Carolina. The papers contained in the volume present a snapshot of rapid and rich developments in the emerging research field known as topological recursion. It has its origin around 2004 in random matrix theory and also in Mirzakhani's work on the volume of moduli spaces of hyperbolic surfaces. Topological recursion has played a fundamental role in connecting seemingly unrelated areas of mathematics such as matrix models, enumeration of Hurwitz numbers and Grothendieck's dessins d'enfants, Gromov-Witten invariants, the A-polynomials and colored polynomial invariants of knots, WKB analysis, and quantization of Hitchin moduli spaces. In addition to establishing these topics, the volume includes survey papers on the most recent key accomplishments: discovery of the unexpected relation to semi-simple cohomological field theories and a solution to the remodeling conjecture. It also provides a glimpse into the future research direction; for example, connections with the Airy structures, modular functors, Hurwitz-Frobenius manifolds, and ELSV-type formulas.

Geometry of Moduli Spaces and Representation Theory

Roman Bezrukavnikov,Alexander Braverman,Zhiwei Yun
Geometry of Moduli Spaces and Representation Theory Book PDF

Author : Roman Bezrukavnikov,Alexander Braverman,Zhiwei Yun
Publisher : American Mathematical Soc.
Page : 436 pages
File Size : 55,6 Mb
Release : 2017-12-15
Category : Algebraic varieties
ISBN : 9781470435745

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Geometry of Moduli Spaces and Representation Theory by Roman Bezrukavnikov,Alexander Braverman,Zhiwei Yun Pdf

This book is based on lectures given at the Graduate Summer School of the 2015 Park City Mathematics Institute program “Geometry of moduli spaces and representation theory”, and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory. Geometric representation theory is a young but fast developing research area at the intersection of these subjects. An early profound achievement was the famous conjecture by Kazhdan–Lusztig about characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson-Bernstein and Brylinski-Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry. Another focus of the program was enumerative algebraic geometry. Recent progress showed the role of Lie theoretic structures in problems such as calculation of quantum cohomology, K-theory, etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections. The main goal of this collection of notes is to provide young researchers and experts alike with an introduction to these areas of active research and promote interaction between the two related directions.

Quantum Mechanics in the Geometry of Space-Time

Roger Boudet
Quantum Mechanics in the Geometry of Space Time Book PDF

Author : Roger Boudet
Publisher : Springer Science & Business Media
Page : 126 pages
File Size : 54,9 Mb
Release : 2011-06-13
Category : Science
ISBN : 9783642191992

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Quantum Mechanics in the Geometry of Space-Time by Roger Boudet Pdf

This book continues the fundamental work of Arnold Sommerfeld and David Hestenes formulating theoretical physics in terms of Minkowski space-time geometry. We see how the standard matrix version of the Dirac equation can be reformulated in terms of a real space-time algebra, thus revealing a geometric meaning for the “number i” in quantum mechanics. Next, it is examined in some detail how electroweak theory can be integrated into the Dirac theory and this way interpreted in terms of space-time geometry. Finally, some implications for quantum electrodynamics are considered. The presentation of real quantum electromagnetism is expressed in an addendum. The book covers both the use of the complex and the real languages and allows the reader acquainted with the first language to make a step by step translation to the second one.

Geometric Complexity Theory IV: Nonstandard Quantum Group for the Kronecker Problem

Jonah Blasiak,Ketan D. Mulmuley,Milind Sohoni
Geometric Complexity Theory IV Nonstandard Quantum Group for the Kronecker Problem Book PDF

Author : Jonah Blasiak,Ketan D. Mulmuley,Milind Sohoni
Publisher : American Mathematical Soc.
Page : 160 pages
File Size : 54,9 Mb
Release : 2015-04-09
Category : Mathematics
ISBN : 9781470410117

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Geometric Complexity Theory IV: Nonstandard Quantum Group for the Kronecker Problem by Jonah Blasiak,Ketan D. Mulmuley,Milind Sohoni Pdf

The Kronecker coefficient is the multiplicity of the -irreducible in the restriction of the -irreducible via the natural map , where are -vector spaces and . A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. The authors construct two quantum objects for this problem, which they call the nonstandard quantum group and nonstandard Hecke algebra. They show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality.

Introduction to Quantum Groups

Masud Chaichian,Andrei Pavlovich Demichev
Introduction to Quantum Groups Book PDF

Author : Masud Chaichian,Andrei Pavlovich Demichev
Publisher : World Scientific
Page : 362 pages
File Size : 49,9 Mb
Release : 1996
Category : Science
ISBN : 9810226233

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Introduction to Quantum Groups by Masud Chaichian,Andrei Pavlovich Demichev Pdf

In the past decade there has been an extemely rapid growth in the interest and development of quantum group theory.This book provides students and researchers with a practical introduction to the principal ideas of quantum groups theory and its applications to quantum mechanical and modern field theory problems. It begins with a review of, and introduction to, the mathematical aspects of quantum deformation of classical groups, Lie algebras and related objects (algebras of functions on spaces, differential and integral calculi). In the subsequent chapters the richness of mathematical structure and power of the quantum deformation methods and non-commutative geometry is illustrated on the different examples starting from the simplest quantum mechanical system — harmonic oscillator and ending with actual problems of modern field theory, such as the attempts to construct lattice-like regularization consistent with space-time Poincaré symmetry and to incorporate Higgs fields in the general geometrical frame of gauge theories. Graduate students and researchers studying the problems of quantum field theory, particle physics and mathematical aspects of quantum symmetries will find the book of interest.

Geometric and Algebraic Topological Methods in Quantum Mechanics

G. Giachetta,L. Mangiarotti,Gennadi? Aleksandrovich Sardanashvili
Geometric and Algebraic Topological Methods in Quantum Mechanics Book PDF

Author : G. Giachetta,L. Mangiarotti,Gennadi? Aleksandrovich Sardanashvili
Publisher : World Scientific
Page : 716 pages
File Size : 48,9 Mb
Release : 2005
Category : Science
ISBN : 9789812561299

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Geometric and Algebraic Topological Methods in Quantum Mechanics by G. Giachetta,L. Mangiarotti,Gennadi? Aleksandrovich Sardanashvili Pdf

- The book collects all the advanced methods of quantization in the last decade. - It presents in a compact way all the necessary up to date mathematical tools to be used in studying quantum problems.

Geometric, Algebraic and Topological Methods for Quantum Field Theory

Alexander Cardona,Carolina Neira-Jiménez,Hernán Ocampo,Sylvie Paycha,Andrés F Reyes-Lega
Geometric Algebraic and Topological Methods for Quantum Field Theory Book PDF

Author : Alexander Cardona,Carolina Neira-Jiménez,Hernán Ocampo,Sylvie Paycha,Andrés F Reyes-Lega
Publisher : World Scientific
Page : 380 pages
File Size : 50,8 Mb
Release : 2013-11-15
Category : Mathematics
ISBN : 9789814460064

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Geometric, Algebraic and Topological Methods for Quantum Field Theory by Alexander Cardona,Carolina Neira-Jiménez,Hernán Ocampo,Sylvie Paycha,Andrés F Reyes-Lega Pdf

Based on lectures held at the 7th Villa de Leyva summer school, this book presents an introduction to topics of current interest in the interface of geometry, topology and physics. It is aimed at graduate students in physics or mathematics with interests in geometric, algebraic as well as topological methods and their applications to quantum field theory. This volume contains the written notes corresponding to lectures given by experts in the field. They cover current topics of research in a way that is suitable for graduate students of mathematics or physics interested in the recent developments and interactions between geometry, topology and physics. The book also contains contributions by younger participants, displaying the ample range of topics treated in the school. A key feature of the present volume is the provision of a pedagogical presentation of rather advanced topics, in a way which is suitable for both mathematicians and physicists. Contents:Lectures:Spectral Geometry (B Iochum)Index Theory for Non-compact G-manifolds (M Braverman and L Cano)Generalized Euler Characteristics, Graph Hypersurfaces, and Feynman Periods (P Aluffi)Gravitation Theory and Chern-Simons Forms (J Zanelli)Noncommutative Geometry Models for Particle Physics (M Marcolli)Noncommutative Spacetimes and Quantum Physics (A P Balachandran)Integrability and the AdS/CFT Correspondence (M Staudacher)Compactifications of String Theory and Generalized Geometry (M Graña and H Triendl)Short Communications:Groupoids and Poisson Sigma Models with Boundary (A Cattaneo and I Contreras)A Survey on Orbifold String Topology (A Angel)Grothendieck Ring Class of Banana and Flower Graphs (P Morales-Almazán)On the Geometry Underlying a Real Lie Algebra Representation (R Vargas Le-Bert) Readership: Researchers in geometry and topology, mathematical physics. Keywords:Geometry;Topology;Geometric Methods;Quantum Field Theory;Renormalization;Index Theory;Noncommutative Geometry;Quantization;String Theory;Key Features:Unique style aimed at a mixed readership of mathematicians and physicistsIdeal for self-study or use in advanced courses or seminars

Strings and Geometry

Clay Mathematics Institute. Summer School,Isaac Newton Institute for Mathematical Sciences
Strings and Geometry Book PDF

Author : Clay Mathematics Institute. Summer School,Isaac Newton Institute for Mathematical Sciences
Publisher : American Mathematical Soc.
Page : 396 pages
File Size : 55,9 Mb
Release : 2004
Category : Mathematics
ISBN : 082183715X

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Strings and Geometry by Clay Mathematics Institute. Summer School,Isaac Newton Institute for Mathematical Sciences Pdf

Contains selection of expository and research article by lecturers at the school. Highlights current interests of researchers working at the interface between string theory and algebraic supergravity, supersymmetry, D-branes, the McKay correspondence andFourer-Mukai transform.

Quantum Stochastic Processes and Noncommutative Geometry

Kalyan B. Sinha,Debashish Goswami
Quantum Stochastic Processes and Noncommutative Geometry Book PDF

Author : Kalyan B. Sinha,Debashish Goswami
Publisher : Cambridge University Press
Page : 301 pages
File Size : 42,7 Mb
Release : 2007-01-25
Category : Mathematics
ISBN : 9781139461696

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Quantum Stochastic Processes and Noncommutative Geometry by Kalyan B. Sinha,Debashish Goswami Pdf

The classical theory of stochastic processes has important applications arising from the need to describe irreversible evolutions in classical mechanics; analogously quantum stochastic processes can be used to model the dynamics of irreversible quantum systems. Noncommutative, i.e. quantum, geometry provides a framework in which quantum stochastic structures can be explored. This book is the first to describe how these two mathematical constructions are related. In particular, key ideas of semigroups and complete positivity are combined to yield quantum dynamical semigroups (QDS). Sinha and Goswami also develop a general theory of Evans-Hudson dilation for both bounded and unbounded coefficients. The unique features of the book, including the interaction of QDS and quantum stochastic calculus with noncommutative geometry and a thorough discussion of this calculus with unbounded coefficients, will make it of interest to graduate students and researchers in functional analysis, probability and mathematical physics.

New Dualities of Supersymmetric Gauge Theories

Jörg Teschner
New Dualities of Supersymmetric Gauge Theories Book PDF

Author : Jörg Teschner
Publisher : Springer
Page : 467 pages
File Size : 42,7 Mb
Release : 2015-11-17
Category : Science
ISBN : 9783319187693

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New Dualities of Supersymmetric Gauge Theories by Jörg Teschner Pdf

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.

Topological Strings and Quantum Curves

Lotte Hollands
Topological Strings and Quantum Curves Book PDF

Author : Lotte Hollands
Publisher : Amsterdam University Press
Page : 310 pages
File Size : 53,6 Mb
Release : 2009
Category : Mathematics
ISBN : 9789085550204

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Topological Strings and Quantum Curves by Lotte Hollands Pdf

This thesis presents several new insights on the interface between mathematics and theoretical physics, with a central role for Riemann surfaces. First of all, the duality between Vafa-Witten theory and WZW models is embedded in string theory. Secondly, this model is generalized to a web of dualities connecting topological string theory and N=2 supersymmetric gauge theories to a configuration of D-branes that intersect over a Riemann surface. This description yields a new perspective on topological string theory in terms of a KP integrable system based on a quantum curve. Thirdly, this thesis describes a geometric analysis of wall-crossing in N=4 string theory. And lastly, it offers a novel approach to constuct metastable vacua in type IIB string theory.