On higher order Sugawara operators

Abstract

The higher Sugawara operators acting on the Verma modules over the affine Kac-Moody algebra at the critical level are related to the higher Hamiltonians of the Gaudin model due to work of Feigin, Frenkel and Reshetikhin. An explicit construction of the higher Hamiltonians in the case of gl_n was given recently by Talalaev. We propose a new approach to these results from the viewpoint of the vertex algebra theory by proving directly the formulas for the higher order Sugawara operators. The eigenvalues of the operators in the Wakimoto modules of critical level are also calculated.

20 pages, some corrections are made, references added

References in corpus (4)

• Vertex Algebras and Algebraic Curves
• Gaudin Model, Bethe Ansatz and Critical Level
• Finite vs. affine W-algebras
• Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence

Cited by in corpus (30)

• Feigin-Frenkel center in types B, C and D
• Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras
• Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases
• Center of the quantum affine vertex algebra in type A
• Higher Sugawara operators for the quantum affine algebras of type A
• Cyclotomic Gaudin models: construction and Bethe ansatz
• Quantization of the shift of argument subalgebras in type A
• The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{gl}}(m|n)$
• Classical W-algebras in types A, B, C, D and G
• Preservation of depth in local geometric Langlands correspondence
• Yangian characters and classical W-algebras
• Invariants of the vacuum module associated with the Lie superalgebra gl(1|1)
• Segal-Sugawara vectors for the Lie algebra of type $G_2$
• Cyclotomic Gaudin models with irregular singularities
• Manin Matrices for Quadratic Algebras
• Eigenvalues of Bethe vectors in the Gaudin model
• Quasi modules for the quantum affine vertex algebra in type $A$
• On Segal--Sugawara vectors and Casimir elements for classical Lie algebras
• A new formula for Pfaffian-type Segal-Sugawara vector
• Center at the critical level for centralizers in type $A$
• Decomplexification of the Capelli identities and holomorphic factorization
• Quantizing Mishchenko-Fomenko subalgebras for centralizers via affine W-algebras
• Pfaffian-type Sugawara operators
• Suzuki functor at the critical level
• Compatibility of the Feigin-Frenkel Isomorphism and the Harish-Chandra Isomorphism for jet algebras
• Quantum Representation Theory and Manin matrices I: finite-dimensional case
• Manin matrices of type C: multi-parametric deformation
• Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras
• Segal-Sugawara vectors for orthosymplectic Lie superalgebras
• Introduction to finite $W$-algebras