Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal \mathfrakgl₂

2025/05/05 by Yusuke Ohkubo, Ohkubo, Yusuke
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2505.02926

openalex publication_date 2025/05/05 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

It is known that the q-deformed Virasoro algebra can be constructed from a certain representation of the quantum toroidal \mathfrakgl₁ algebra. In this paper, we apply the same construction to the quantum toroidal algebra of type \mathfrakgl₂ and study the properties of resulting generators W_i(z) (i=1,2). The algebra generated by W_i(z) can be regarded as a q-deformation of the direct sum F ⊕ SVir, where F denotes the free fermion algebra and SVir stands for the N=1 super Virasoro algebra, also referred to as the N=1 superconformal algebra or the Neveu-Schwarz-Ramond algebra. Moreover, the generators W_i(z) admit two screening currents, and we show that their degeneration limits coincide with the screening currents of SVir. We also establish quadratic relations satisfied by W_i(z) and show that they generate a pair of commuting q-deformed Virasoro algebras, which degenerate into two nontrivial commuting Virasoro algebras included in F ⊕ SVir.

Citations

• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• Deformations of \mathcal W algebras via quantum toroidal algebras
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• Vertex Algebras at the Corner
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• Liouville Correlation Functions from Four-Dimensional Gauge Theories
• Quantum toroidal algebras and their vertex representations
• Generalization and Deformation of Drinfeld quantum affine algebras
• SUPERCONFORMAL 2D MINIMAL MODELS AND AN UNUSUAL COSET CONSTRUCTION

Related

• A (q,t)-deformation of the 2d Toda integrable hierarchy
• Superintegrability of q,t-matrix models and quantum toroidal algebra recursions
• Deformations of \mathcal W algebras via quantum toroidal algebras
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• An integral form of quantum toroidal \mathfrakgl₁