Mathematics

Given a star product with separation of variables ⋆ on a pseudo-Kähler manifold M and a point x 0 ∈M , we construct an associative algebra of formal distributions supported at x 0 . We use this algebra to express the formal oscillatory exponents of a family of formal oscillatory integrals related to the star product ⋆ .

We prove the existence of a homomorphism of vertex algebras, from the vacuum Verma module over the loop algebra of an untwisted affine algebra, whose construction is analogous to that of the Feigin-Frenkel homomorphism from the vacuum Verma module at critical level over an affine algebra.

We aim to explore if inside a quantum vertex algebras, we can find the right notion of a quantum conformal algebra.

It is known that the category C H of nilpotent weight modules over the quantum group associated with the super Lie algebra sl(2|1) is a relative pre-modular G -category. Its modified trace enables to define an invariant of 3 -manifolds. In this article we show that the category C H is a relative modular G -category which allows one to construct a family of non-semi-simple extended topological quantum field theories which surprisingly are anomaly free. The quantum group associated with sl(2|1) is considered at odd roots of unity.

We introduce an associative algebra A ∞ (V) using infinite matrices with entries in a grading-restricted vertex algebra V such that the associated graded space Gr(W)= ∐ n∈N G r n (W) of a filtration of a lower-bounded generalized V -module W is an A ∞ (V) -module satisfying additional properties (called a graded A ∞ (V) -module). We prove that a lower-bounded generalized V -module W is irreducible or completely reducible if and only if the graded A ∞ (V) -module Gr(W) is irreducible or completely reducible, respectively. We also prove that the set of equivalence classes of the lower-bounded generalized V -modules are in bijection with the set of the equivalence classes of graded A ∞ (V) -modules. For N∈N , there is a subalgebra A N (V) of A ∞ (V) such that the subspace G r N (W)= ∐ N n=0 G r n (W) of Gr(W) is an A N (V) -module satisfying additional properties (called a graded A N (V) -module). We prove that A N (V) are finite dimensional when V is of positive energy (CFT type) and C 2 -cofinite. We prove that the set of the equivalence classes of lower-bounded generalized V -modules is in bijection with the set of the equivalence classes of graded A N (V) -modules. In the case that V is a Möbius vertex algebra and the differences between the real parts of the lowest weights of the irreducible lower-bounded generalized V -modules are less than or equal to N∈N , we prove that a lower-bounded generalized V -module W of finite length is irreducible or completely reducible if and only if the graded A N (V) -module G r N (W) is irreducible or completely reducible, respectively.

We compute the monoidal and braided auto-equivalences of the modular tensor categories C( sl r+1 ,k) , C( so 2r+1 ,k) , C( sp 2r ,k) , and C( g 2 ,k) . Along with the expected simple current auto-equivalences, we show the existence of the charge conjugation auto-equivalence of C( sl r+1 ,k) , and exceptional auto-equivalences of C( so 2r+1 ,2) , C( sp 2r ,r) , C( g 2 ,4) . We end the paper with a section discussing potential applications of these computations.

We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra V L ( ℓ 123 ,0) .Then, for any integer t>1 , we introduce a new Lie algebra L t , and show that σ t -twisted V L ( ℓ 123 ,0) ( ℓ 2 =0 )-modules are in one-to-one correspondence with restricted L t -modules of level ℓ 13 , where σ t is an order t automorphism of V L ( ℓ 123 ,0) . At the end, we give a complete list of irreducible σ t -twisted V L ( ℓ 123 ,0) ( ℓ 2 =0 )-modules.

We have reviewed some results on quantized shuffling, and in particular, the grading and structure of this algebra. In parallel, we have summarized certain details about classical shuffle algebras, including Lyndon words (primes) and the construction of bases of classical shuffle algebras in terms of Lyndon words. We have explained how to adapt this theory to the construction of bases of quantum group algebras in terms of Lyndon words. This method has a limited application to the specific case of the quantum group parameter being a root of unity, with the requirement that specialization to the root of unity is non-restricted. As an additional, applied part of this work, we have implemented a Wolfram Mathematica package with functions for quantum shuffle multiplication and constructions of bases in terms of Lyndon words.

This paper continues an earlier research of the authors on universal quadratic identities (QIs) on minors of quantum matrices. We demonstrate situations when the universal QIs are provided, in a sense, by the ones of four special types (Plucker, co-Plucker, Dodgson identities and quasi-commutation relations on flag and co-flag interval minors).

We study the simple Bershadsky-Polyakov algebra W k = W k (s l 3 , f θ ) at positive integer levels and classify their irreducible modules. In this way we confirm the conjecture from arXiv:1910.13781. Next, we study the case k=1 . We discover that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple afine vertex superalgebra L k ′ (osp(1|2)) for k ′ =−5/4 . Using the free-field realization of L k ′ (osp(1|2)) from arXiv:1711.11342, we get a free-field realization of W k and their highest weight modules. In a sequel, we plan to study fusion rules for W k .