182
D. Valeri
representation theory of superconformal algebras. Quantum affine W -algebras may
be also considered as an affinization of quantum finite W -algebras [46] which are a
natural quantization of Slodowy slices [34].
W -algebras are at the cross roads of representation theory and mathematical
physics and play important roles (just to cite some of them) in applications to
integrable systems [15, 24], to Gromov–Witten theory and singularity theory [3, 44],
the geometric Langlands program [28, 31–33], four-dimensional gauge theories
[1, 8, 48].
In this note we survey the recent approach to (quantum finite and classical affine)
W -algebras based on the notion of Lax type operators [18–21]. For a review of the
approach to (classical) W -algebras via generators and relations we refer to [13].
Throughout the paper the base field F is a field of characteristic zero.
2 What Is a W -Algebra?
W -algebras are a rich family of algebraic structures associated to a pair (g, f )
consisting of a finite-dimensional reductive Lie algebra g and a nilpotent element
f ∈ g. They are obtained via Hamiltonian reduction in different categories: Poisson
algebras, associative algebras and (Poisson), vertex algebras. We should think of
them as algebraic structures underlying some physical theories with “extended
symmetries.”
2.1 Fundamental Physical Theories and Corresponding
Fundamental Algebraic Structures
In Classical Mechanics the phase space, describing the possible configurations of
a physical system, is a Poisson manifold. The physical observables are the smooth
functions on the manifold, and they thus form a Poisson algebra (PA).
By quantizing this theory we go to Quantum Mechanics. The observables
become noncommutative objects, elements of an associative algebra (AA). Hence,
the Poisson bracket is replaced by the usual commutator and the phase space is
described as a representation of this associative algebra.
Going from a finite to an infinite number of degrees of freedom, we pass
from Classical and Quantum Mechanics to Classical and Quantum Field Theory,
respectively. The algebraic structure corresponding to an arbitrary Quantum Field
Theory is still to be understood, but in the special case of chiral quantum fields of
a 2-dimensional Conformal Field Theory (CFT) the adequate algebraic structure is
a vertex algebra (VA) [6], and its quasi-classical limit is known as Poisson vertex
algebra (PVA) [14].
D. Valeri
representation theory of superconformal algebras. Quantum affine W -algebras may
be also considered as an affinization of quantum finite W -algebras [46] which are a
natural quantization of Slodowy slices [34].
W -algebras are at the cross roads of representation theory and mathematical
physics and play important roles (just to cite some of them) in applications to
integrable systems [15, 24], to Gromov–Witten theory and singularity theory [3, 44],
the geometric Langlands program [28, 31–33], four-dimensional gauge theories
[1, 8, 48].
In this note we survey the recent approach to (quantum finite and classical affine)
W -algebras based on the notion of Lax type operators [18–21]. For a review of the
approach to (classical) W -algebras via generators and relations we refer to [13].
Throughout the paper the base field F is a field of characteristic zero.
2 What Is a W -Algebra?
W -algebras are a rich family of algebraic structures associated to a pair (g, f )
consisting of a finite-dimensional reductive Lie algebra g and a nilpotent element
f ∈ g. They are obtained via Hamiltonian reduction in different categories: Poisson
algebras, associative algebras and (Poisson), vertex algebras. We should think of
them as algebraic structures underlying some physical theories with “extended
symmetries.”
2.1 Fundamental Physical Theories and Corresponding
Fundamental Algebraic Structures
In Classical Mechanics the phase space, describing the possible configurations of
a physical system, is a Poisson manifold. The physical observables are the smooth
functions on the manifold, and they thus form a Poisson algebra (PA).
By quantizing this theory we go to Quantum Mechanics. The observables
become noncommutative objects, elements of an associative algebra (AA). Hence,
the Poisson bracket is replaced by the usual commutator and the phase space is
described as a representation of this associative algebra.
Going from a finite to an infinite number of degrees of freedom, we pass
from Classical and Quantum Mechanics to Classical and Quantum Field Theory,
respectively. The algebraic structure corresponding to an arbitrary Quantum Field
Theory is still to be understood, but in the special case of chiral quantum fields of
a 2-dimensional Conformal Field Theory (CFT) the adequate algebraic structure is
a vertex algebra (VA) [6], and its quasi-classical limit is known as Poisson vertex
algebra (PVA) [14].