[Paper Review] Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator

Koichi Harada|arXiv (Cornell University)|May 28, 2020

Algebraic structures and combinatorial models61 references4 citations

TL;DR

This paper constructs the quantum deformation of Feigin-Semikhatov's W-algebras associated with the $υ(N-1,1)$ $υ(\mathfrak{su}(2)$ embedding via gluing two quantum toroidal algebras of $υ(\mathfrak{gl}_1$. It establishes free field realizations, derives quadratic relations, and confirms the match between the norm of Whittaker states and the 5d instanton partition function with a simple surface operator in the $N=3$ case, providing evidence for the 5d AGT correspondence.

ABSTRACT

The quantum toroidal algebra of $gl_1$ provides many deformed W-algebras associated with (super) Lie algebras of type A. The recent work by Gaiotto and Rapcak suggests that a wider class of deformed W-algebras including non-principal cases are obtained by gluing the quantum toroidal algebras of $gl_1$. These algebras are expected to be related with 5d AGT correspondence. In this paper, we discuss quantum deformation of the W-algebras obtained from $\widehat{su}(N)$ by the quantum Drinfeld-Sokolov reduction with su(2) embedding [N-1,1]. They were studied by Feigin and Semikhatov and we refer to them as Feigin-Semikhatov's W-algebras. We construct free field realization and find several quadratic relations. We also compare the norm of the Whittaker states with the instanton partition function under the presence of a simple surface operator in the N=3 case.

Motivation & Objective

• To construct a quantum deformation of Feigin-Semikhatov's W-algebras for the $υ(N-1,1)$ $υ(\mathfrak{su}(2)$ embedding using quantum toroidal algebras.
• To extend Gaiotto-Rapčák’s framework to the q-deformed case by gluing two quantum toroidal algebras of $υ(\mathfrak{gl}_1$.
• To establish free field realizations and derive quadratic relations for the deformed W-algebras.
• To compare the norm of Whittaker states with the 5d instanton partition function in the presence of a simple surface operator.
• To provide evidence for the 5d AGT correspondence through this comparison in the $N=3$ case.

Proposed method

• Utilizes the quantum toroidal algebra of $υ(\mathfrak{gl}_1$ (Ding-Iohara-Miki algebra) as a unifying framework for deformed W-algebras.
• Constructs deformed vertex operators by decomposing $U_q(\widehat{\mathfrak{gl}}_2)$ into two quantum toroidal algebras of $υ(\mathfrak{gl}_1$, generalizing from the $N=2$ case.
• Applies the gluing construction to lift Gaiotto-Rapčák’s Y-algebra framework to the q-deformed setting.
• Derives quadratic relations for the deformed currents $\mathscr{E}'_N(z)$ using operator product expansions and braiding factors involving $f(w/z)$.
• Uses induction and delta-function identities to prove the quadratic relation (4.46) for general $N$, relying on the $N$-case assumption.
• Compares the norm of Whittaker states with the 5d instanton partition function via character matching and state normalization.

Experimental results

Research questions

• RQ1 How can Feigin-Semikhatov’s W-algebras for the $[N-1,1]$ $υ(\mathfrak{su}(2)$ embedding be quantum-deformed using quantum toroidal algebras of $υ(\mathfrak{gl}_1$?
• RQ2 Can the gluing construction of quantum toroidal algebras reproduce the deformed W-algebra structure and its free field realization?
• RQ3 What is the explicit form of the quadratic relations satisfied by the deformed currents in the $N$-case?
• RQ4 Does the norm of Whittaker states in the deformed W-algebra match the 5d instanton partition function with a simple surface operator?
• RQ5 How does the $N=3$ case confirm the 5d AGT correspondence through this matching?

Key findings

• The quantum deformation of Feigin-Semikhatov’s W-algebra is successfully constructed via gluing two quantum toroidal algebras of $υ(\mathfrak{gl}_1$, generalizing the $N=2$ case.
• Free field realizations are established for the deformed W-algebras, providing a concrete representation in terms of bosonic fields.
• Quadratic relations for the deformed currents $\mathscr{E}'_N(z)$ are derived and proven by induction, with the key relation (4.46) holding for all $N$.
• The norm of Whittaker states in the $N=3$ case matches exactly with the 5d instanton partition function under the presence of a simple surface operator.
• The proof relies on braiding factors $f(w/z)$ and delta-function identities to cancel cross-terms, confirming the consistency of the algebraic structure.
• The results provide strong evidence for the 5d AGT correspondence in the presence of surface operators through the matching of algebraic and physical partition functions.