QFT, STRINGS AND LOW-DIMENSIONAL TOPOLOGY

2002 ◽  
Vol 17 (35) ◽  
pp. 2289-2295
Author(s):  
HU SEN
Keyword(s):  
Mirror Symmetry ◽  
Theoretical Physics ◽  
Yang Mills ◽  
Low Dimensional Topology ◽  
Witten Theory ◽  
Close Relationship ◽  
Jones Polynomials ◽  
Recent Developments ◽  
Low Dimensional ◽  
Chern Simons

In between the 80's and 90's we witnessed deep interactions between mathematics and theoretical physics, especially in the understanding of low-dimensional topology in terms of quantum field theory. For example, Jones polynomials (Chern–Simons–Witten theory), Donaldson and Seiberg–Witten invariants (SUSY Yang–Mills theory) and mirror symmetry (T duality in strings) are all naturally understood in terms of QFT and strings. Recent developments indicate a close relationship between gauge theory and gravity theory both in physics and in low-dimensional topology. We shall survey these developments and report some of our work. We shall also find that the keys to connect geometric and physical objects are through symmetry and quantization.

Introduction

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Mirror Symmetry ◽  
Complex Geometry ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Hamiltonian Formalism ◽  
Theoretical Physics ◽  
Lagrange Equations ◽  
Low Dimensional Topology ◽  
The Subject ◽  
Low Dimensional

Symplectic topology has a long history. It has its roots in classical mechanics and geometric optics and in its modern guise has many connections to other fields of mathematics and theoretical physics ranging from dynamical systems, low-dimensional topology, algebraic and complex geometry, representation theory, and homological algebra, to classical and quantum mechanics, string theory, and mirror symmetry. One of the origins of the subject is the study of the equations of motion arising from the Euler–Lagrange equations of a one-dimensional variational problem. The Hamiltonian formalism arising from a Legendre transformation leads to the notion of a ...

Geometry and Physics: Volume II

2018 ◽  
Keyword(s):  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Geometric Analysis ◽  
Mathematical Physics ◽  
Poisson Geometry ◽  
Special Holonomy ◽  
Low Dimensional Topology ◽  
Generalized Complex Structures ◽  
Wide Range ◽  
Low Dimensional

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

Geometry and Physics: Volume I

2018 ◽  
Keyword(s):  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Geometric Analysis ◽  
Mathematical Physics ◽  
Poisson Geometry ◽  
Special Holonomy ◽  
Low Dimensional Topology ◽  
Generalized Complex Structures ◽  
Wide Range ◽  
Low Dimensional

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

scholarly journals Introduction to Chern-Simons Theory

2020 ◽  
Author(s):  
Adémọ́lá Adéìfẹ́ọba
Keyword(s):  
Quantum Hall ◽  
Simons Theory ◽  
Yang Mills ◽  
Jones Polynomials ◽  
Interaction Term ◽  
Quantum Hall System ◽  
Chern Simons ◽  
Theoretic Description ◽  
Chern Simons Theory ◽  
Topological Term

The 2 + 1 Yang-Mills theory allows for an interaction term called the Chern-Simons term. This topological term plays a useful role in understanding the field theoretic description of the excitation of the quantum hall system such as Anyons. While solving the non-Abelian Chern-simons theory is rather complicated, its knotty world allows for a framework for solving it. In the framework, the idea was to relate physical observables with the Jones polynomials. In this note, I will summarize the basic idea leading up to this framework.

DOES HETEROTIC STRING GENERATE CHERN-SIMONS ACTION IN THREE DIMENSIONS?

1992 ◽  
Vol 07 (20) ◽  
pp. 1805-1815 ◽  
Author(s):  
HITOSHI NISHINO
Keyword(s):  
Field Strength ◽  
String Tension ◽  
Heterotic String ◽  
Three Dimensions ◽  
Simons Theory ◽  
Coupling Constant ◽  
Yang Mills ◽  
Close Relationship ◽  
Chern Simons ◽  
Chern Simons Theory

We study the possibility that the ten-dimensional (D=10) heterotic superstring generates a class of Chern-Simons theories in three dimensions, via compactifications on the internal seven-dimensional manifolds. We give an explicit example of such compactifications on (Calabi-Yau)6×S1×(Mink.)3, using the dual formulation of the heterotic string. The string tension 1/α′ as well as the Yang-Mills coupling constant g2 in D=10 is double-quantized in terms of two integers, through a condensate of the antisymmetric field strength NM1…M7, which is dual to the ordinary field strength GMNP. The resultant Chern-Simons theory has the gauge group E6⊗E8, containing also the usual kinetic terms of the gauge fields, which may be interpreted as regulator terms. This result suggests a close relationship between the D=10 heterotic string and D=3 Chern-Simons theory.

scholarly journals Quantum automata, braid group and link polynomials

2007 ◽  
Vol 7 (5&6) ◽  
pp. 479-503
Author(s):  
S. Garnerone ◽  
A. Marzuoli ◽  
M. Rasetti
Keyword(s):  
Braid Group ◽  
Theoretical Background ◽  
Simons Theory ◽  
Tensor Algebra ◽  
Spin Network ◽  
Quantum Automata ◽  
Low Dimensional Topology ◽  
Root Of Unity ◽  
Jones Polynomials ◽  
Low Dimensional

The spin--network quantum simulator model, which essentially encodes the (quantum deformed) $SU(2)$ Racah--Wigner tensor algebra, is particularly suitable to address problems arising in low dimensional topology and group theory. In this combinatorial framework we implement families of finite--states and discrete--time quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are colored Jones polynomials. The automaton calculation of the polynomial of (the plat closure of) a link $L$ on $2N$ strands at any fixed root of unity is shown to be bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index $2N$, on the other. The growth rate of the time complexity function in terms of the integer $k$ appearing in the root of unity $q$ can be estimated to be (polynomially) bounded by resorting to the field theoretical background given by the Chern--Simons theory.

scholarly journals Stochastic Processes and Random Matrices

2018 ◽  
Keyword(s):  
Stochastic Processes ◽  
Random Matrices ◽  
Matrix Theory ◽  
Fundamental Problem ◽  
Universality Class ◽  
Theoretical Physics ◽  
The Past ◽  
Recent Developments ◽  
Long Time ◽  
Low Dimensional

The field of stochastic processes and random matrix theory (RMT) has been a rapidly evolving subject during the past fifteen years where the continuous development and discovery of new tools, connections, and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar–Parisi–Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the past twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensemble of random matrices. These chapters not only cover this topic in detail but also present more recent developments that have emerged from these discoveries, for instance in the context of low-dimensional heat transport (on the physics side) or in the context of integrable probability (on the mathematical side).

scholarly journals Cohomological localization of $$ \mathcal{N} $$ = 2 gauge theories with matter

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Guido Festuccia ◽  
Anastasios Gorantis ◽  
Antonio Pittelli ◽  
Konstantina Polydorou ◽  
Lorenzo Ruggeri
Keyword(s):  
Vector Field ◽  
Extended Supersymmetry ◽  
General Framework ◽  
Gauge Theories ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Explicit Result ◽  
Yang Mills ◽  
Self Duality ◽  
Witten Theory

Abstract We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general $$ \mathcal{N} $$ N = 2 gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun’s theory on S4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.

Infrared finiteness of the two-loop correction to the Chern–Simons term in the Yang–Mills–Chern–Simons theory

1995 ◽  
Vol 73 (5-6) ◽  
pp. 344-348 ◽  
Author(s):  
Yeong-Chuan Kao ◽  
Hsiang-Nan Li
Keyword(s):  
Effective Action ◽  
Background Field ◽  
Landau Gauge ◽  
Quantum Corrections ◽  
Simons Theory ◽  
Loop Correction ◽  
Loop Contribution ◽  
Yang Mills ◽  
Chern Simons ◽  
Chern Simons Theory

We show that the two-loop contribution to the coefficient of the Chern–Simons term in the effective action of the Yang–Mills–Chern–Simons theory is infrared finite in the background field Landau gauge. We also discuss the difficulties in verifying the conjecture, due to topological considerations, that there are no more quantum corrections to the Chern–Simons term other than the well-known one-loop shift of the coefficient.

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