scholarly journals AN INTRODUCTION TO YANGIAN SYMMETRIES

1993 ◽  
Vol 07 (20n21) ◽  
pp. 3517-3530 ◽  
Author(s):  
DENIS BERNARD
Keyword(s):  
Lie Groups ◽  
Transfer Matrix ◽  
Heisenberg Model ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Heisenberg Chain ◽  
Classical Heisenberg Model ◽  
Poisson Lie Groups

We review some aspects of the quantum Yangians as symmetry algebras of two-dimensional quantum field theories. The plan of these notes is the following: 1 — The classical Heisenberg model: •Non-Abelian symmetries; •The generators of the symmetries and the semi-classical Yangians; •An alternative presentation of the semi-classical Yangians; (Digression on Poisson-Lie groups. 2 — The quantum Heisenberg chain: •Non-Abelian symmetries and the quantum Yangians; •The transfer matrix and an alternative presentation of the Yangians.

Integrable Quantum Field Theories

2020 ◽  
pp. 575-621
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Conformal Theory ◽  
Integrable Quantum ◽  
Set Up ◽  
Integrable Quantum Field Theory

Chapter 16 covers the general properties of the integrable quantum field theories, including how an integrable quantum field theory is characterized by an infinite number of conserved charges. These theories are illustrated by means of significant examples, such as the Sine–Gordon model or the Toda field theories based on the simple roots of a Lie algebra. For the deformations of a conformal theory, it shown how to set up an efficient counting algorithm to prove the integrability of the corresponding model. The chapter focuses on two-dimensional models, and uses the term ‘two-dimensional’ to denote both a generic two-dimensional quantum field theory as well as its Euclidean version.

TWO-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES AND FROBENIUS ALGEBRAS

1996 ◽  
Vol 05 (05) ◽  
pp. 569-587 ◽  
Author(s):  
LOWELL ABRAMS
Keyword(s):  
Field Theories ◽  
Algebra Structure ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Direct Sums ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Simple Demonstration ◽  
Topological Quantum Field Theories

We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either “annihilator algebras” — algebras whose socle is a principal ideal — or field extensions. The relationship between two-dimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable two-dimensional topological quantum field theories.

scholarly journals DIFF(Σ) AND METRICS FROM HAMILTONIAN-TQFT'S IN 2+1 DIMENSIONS

1993 ◽  
Vol 08 (24) ◽  
pp. 2277-2283 ◽  
Author(s):  
ROGER BROOKS
Keyword(s):  
Topological Field Theories ◽  
Gauge Theories ◽  
Dimensional Space ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Canonical Gravity ◽  
Topological Quantum ◽  
Topological Field

The constraints of BF topological gauge theories are used to construct Hamiltonians which are anti-commutators of the BRST and anti-BRST operators. Such Hamiltonians are a signature of topological quantum field theories (TQFTs). By construction, both classes of topological field theories share the same phase spaces and constraints. We find that, for (2+1)- and (1+1)-dimensional space-times foliated as M=Σ × ℝ, a homomorphism exists between the constraint algebras of our TQFT and those of canonical gravity. The metrics on the two-dimensional hypersurfaces are also obtained.

scholarly journals STATE SUM CONSTRUCTION OF TWO-DIMENSIONAL OPEN-CLOSED TOPOLOGICAL QUANTUM FIELD THEORIES

2007 ◽  
Vol 16 (09) ◽  
pp. 1121-1163 ◽  
Author(s):  
AARON D. LAUDA ◽  
HENDRYK PFEIFFER
Keyword(s):  
The State ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Topological Quantum ◽  
Finite Set ◽  
Manifolds With Corners ◽  
The Relationship ◽  
Topological Quantum Field Theories

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.

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