特定助教 Ana Kontrec (Representation theory)
My main research focus is in the ﬁeld of vertex algebra theory, and especially W-algebras.
In the study of conformal ﬁeld theory and string theory, vertex operator algebras (VOAs) playa fundamental role as the mathematical foundation for the concept of chiral algebras. Recentdevelopments show connection between four dimensional N = 2 superconformal theories andVOAs as their algebraic invariants. One of the most important families of vertex algebras areaﬃne vertex algebras and their associated W-algebras, which are connected to various aspects ofgeometry and physics, such as quantum Langlands correspondence and symmetries of quantum
ﬁeld theories.
Let g be a simple Lie algebra, f a nilpotent element and k a complex number. The universal
aﬃne W-algebra W k (g, f ) is associated to the aﬃne vertex algebra V k (g) through a certain
homological construction known as quantum Hamiltonian reduction (introduced by Feigin andFrenkel for principal nilpotent element and generalized by Kac, Roan and Wakimoto for arbitrary nilpotent element). Structure and representation theory of a given W-algebra depends
largely on the type of nilpotent element used in the construction, e.g. explicit formulae forgenerators and relations are not known generally, but only in the case of the minimal nilpotent
element f .
One of the simplest examples of W-algebras is the Bershadsky-Polyakov vertex algebra Wk =Wk (sl3 , fθ ), associated to g = sl(3) and the minimal nilpotent element fθ . T. Arakawa provedthat Wk (sl3 , fθ ) is rational for half-integer levels k, while in other cases it is a non-rationalvertex algebra. More recently, for k admissible and non-integral, irreducible Wk –modules wereclassiﬁed by D. Adamović and me in [1] for the boundary cases, and by Fehily, Kawasetsu andRidout in full generality. They showed that every irreducible highest weight module for Wk
is obtained as an image of the admissible modules for Lk (sl(3)) (which are classiﬁed by T.
Arakawa). However, when we pass to integral k, these methods are no longer applicable, sincein this case quantum hamiltonian reduction sends Lk (sl(3))–modules to zero.
In [2], Adamović and I were able to obtain a classiﬁcation of irreducible positive energy
modules L(x, y) for Wk at integer levels, parametrizing the highest weights (x, y) as zeroes ofcertain polynomial functions (derived from the associated Zhu algebra). This is one of the ﬁrst
classiﬁcation results of this type in literature for non-admissible, integral level aﬃne W-algebras.
Another topic which has received a great deal of attention lately is the notion of duality ofW-algebras. Let g be a simple Lie algebra, and L g the Langlands dual Lie algebra of g. If f is aprincipal nilpotent element, the Feigin-Frenkel duality states that W k (g, f ) ≃ W l (L g, f ), wherek ∈ C is non-critical and l is the dual level. However, this type of symmetry does not hold ingeneral for non-principal nilpotent elements, which leads us to the notion of Kazama-Suzuki
type duality. Here the Langlands dual Lie algebra is replaced by a Lie superalgebra. The
notion of Kazama-Suzuki dual was ﬁrst introduced in the context of the duality of the N = 2bsuperconformal algebra and aﬃne Lie algebra sl(2),and later generalized for some other aﬃnevertex algebras and W–algebras.
Assume that U, V are vertex superalgebras. We say that V is the Kazama-Suzuki dual of U if
there exist injective homomorphisms of vertex superalgebras φ1 : V → U ⊗F, φ2 : U → V ⊗F−1 ,so thatV ∼= Com H1 , U ⊗ F , U ∼= Com H2 , V ⊗ F−1 ,1where H1 (resp. H2 ) is a rank one Heisenberg vertex subalgebra of U ⊗ F (resp. V ⊗ F−1 ), F is
Cliﬀord vertex superalgebra and F−1 is a lattice vertex superalgebra associated to the negative√deﬁnite lattice Z −1. In [2], D. Adamović and I showed that the aﬃne vertex superalgebra
Lk′ (osp(1|2)) is the Kazama-Suzuki dual of the Bershadsky-Polyakov algebra Wk (sl3 , fθ ) fork = 1. Relaxed modules for Lk′ (osp(1|2)) are mapped to the ordinary Wk –modules, for which
one expects it is easier to obtain the tensor category structure and fusion rules. As a consequenceof this duality, we were able to obtain a free-ﬁeld realization of Wk and their highest weightmodules.

参考文献[1] D. Adamović, A. Kontrec, Classiﬁcation of irreducible modules for Bershadsky-Polyakov
algebra at certain levels, J. Algebra Appl., Vol. 20 (2021), no. 6, 2150102[2] D. Adamović, A. Kontrec, Bershadsky-Polyakov vertex algebras at positive integer levels andduality, Transformation Groups (2022), https://doi.org/10.1007/s00031-022-09721-z

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