[Paper Review] Free Boson Realization of $U_q(\widehat{sl_N})$

Hidetoshi Awata, Satoru Odake+1 more|arXiv (Cornell University)|May 26, 1993

Algebraic structures and combinatorial models19 citations

TL;DR

This paper constructs a free boson realization of the quantum affine algebra $U_q(\widehat{sl_N})$ at arbitrary level $k$, using a $q$-deformed Heisenberg realization and OPE techniques. The construction generalizes the Wakimoto realization in the $q \to 1$ limit and provides explicit expressions for screening currents and vertex operators that commute with the algebra and create highest weight states, respectively.

ABSTRACT

We construct a realization of the quantum affine algebra $U_q(\widehat{sl_N})$ of an arbitrary level $k$ in terms of free boson fields. In the $q\! ightarrow\! 1$ limit this realization becomes the Wakimoto realization of $\widehat{sl_N}$. The screening currents and the vertex operators(primary fields) are also constructed; the former commutes with $U_q(\widehat{sl_N})$ modulo total difference, and the latter creates the $U_q(\widehat{sl_N})$ highest weight state from the vacuum state of the boson Fock space.

Motivation & Objective

• To extend free field realizations of quantum affine algebras beyond rank 1 and 2 to general $N$.
• To provide a systematic construction of $U_q(\widehat{sl_N})$ in terms of free bosons for arbitrary level $k$, filling a gap in the literature.
• To generalize the Wakimoto realization to the quantum affine case via $q$-deformation.
• To construct screening currents and vertex operators (primary fields) that are essential for computing correlation functions in $q$-deformed conformal field theories.
• To establish a framework for studying higher-rank integrable models such as the $XXZ$ spin chain and $q$-deformed WZNW models.

Proposed method

• Affinization of the $q$-difference operator realization (Heisenberg realization) of $U_q(sl_N)$ to construct $U_q(\widehat{sl_N})$.
• Use of operator product expansion (OPE) techniques to verify algebraic relations and ensure consistency of the realization.
• Definition of screening currents as $q$-deformed exponential operators that commute with $U_q(\widehat{sl_N})$ modulo total differences.
• Construction of vertex operators as $q$-exponentials that create highest weight states from the bosonic vacuum.
• Employment of Jackson integrals and $q$-difference operators to handle $q$-deformed commutators and normal ordering.
• Explicit cancellation of poles in OPEs between currents, screening currents, and vertex operators via pairing of terms with opposite singularities.

Experimental results

Research questions

• RQ1 How can the quantum affine algebra $U_q(\widehat{sl_N})$ be realized in terms of free boson fields for arbitrary $N$ and level $k$?
• RQ2 What is the $q$-deformed analog of the Wakimoto realization for $\widehat{sl_N}$, and how does it reduce to the classical case as $q \to 1$?
• RQ3 How can screening currents be constructed such that they commute with $U_q(\widehat{sl_N})$ modulo total differences?
• RQ4 What is the structure of vertex operators (primary fields) that create highest weight representations from the bosonic vacuum?
• RQ5 How do the OPEs between currents, screening currents, and vertex operators behave, and what ensures their consistency through pole cancellation?

Key findings

• A free boson realization of $U_q(\widehat{sl_N})$ is constructed at arbitrary level $k$, generalizing previous results for $N=2,3$.
• The realization reduces to the classical Wakimoto realization in the $q \to 1$ limit, confirming consistency with known classical constructions.
• Screening currents are explicitly constructed as $q$-exponentials of bosonic fields, and they commute with $U_q(\widehat{sl_N})$ modulo total differences.
• Vertex operators are defined as $q$-exponentials that create $U_q(\widehat{sl_N})$-highest weight states from the bosonic Fock vacuum.
• Pole cancellations in OPEs between currents and screening/vertex operators are systematically verified through pairing of terms with opposite singularities.
• The grading operator is also bosonized, enabling a full algebraic structure within the free field framework.