The N = 1 super Heisenberg–Virasoro vertex algebra at level zero

We study the representation theory of the [Formula: see text] super Heisenberg–Virasoro vertex algebra at level zero, which extends the previous work on the Heisenberg–Virasoro vertex algebra [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342; D. Adamović and G. Radobolja, Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero, Commun. Contemp. Math. 21(2) (2019) 1850008; Y. Billig, Representations of the twisted Heisenberg–Virasoro algebra at level zero, Can. Math. Bull. 46(4) (2003) 529–537] to the super case. We calculated all characters of irreducible highest weight representations by investigating certain Fock space representations. Quite surprisingly, we found that the maximal submodules of certain Verma modules are generated by subsingular vectors. The formulas for singular and subsingular vectors are obtained using screening operators appearing in a study of certain logarithmic vertex algebras [D. Adamović and A. Milas, On W-algebras associated to [Formula: see text] minimal models and their representations, Int. Math. Res. Notices 2010(20) (2010) 3896–3934].

On deformations and extensions of Diff(S2)

We investigate the algebra of vector fields on the sphere. First, we find that linear deformations of this algebra are obstructed under reasonable conditions. In particular, we show that hs[λ], the one-parameter deformation of the algebra of area-preserving vector fields, does not extend to the entire algebra. Next, we study some non-central extensions through the embedding of $$ \mathfrak{vect} $$ vect (S2) into $$ \mathfrak{vect} $$ vect (ℂ*). For the latter, we discuss a three parameter family of non-central extensions which contains the symmetry algebra of asymptotically flat and asymptotically Friedmann spacetimes at future null infinity, admitting a simple free field realization.

Higher genus correlators for tensionless AdS3 strings

It was recently shown in [1] that tree-level correlation functions in tensionless string theory on AdS3 × S3 × $$ {\mathbbm{T}}^4 $$ T 4 match the expected form of correlation functions in the symmetric orbifold CFT on $$ {\mathbbm{T}}^4 $$ T 4 in the large N limit. This analysis utilized the free-field realization of the $$ \mathfrak{psu}{\left(1,\left.1\right|2\right)}_1 $$ psu 1 1 2 1 Wess-Zumino-Witten model, along with a surprising identity directly relating these correlation functions to a branched covering of the boundary of AdS3. In particular, this identity implied the unusual feature that the string theory correlators localize to points in the moduli space for which the worldsheet covers the boundary of AdS3 with specified branching near the insertion points. In this work we generalize this analysis past the tree-level approximation, demonstrating its validity to higher genus worldsheets, and in turn providing strong evidence for this incarnation of the AdS/CFT correspondence at all orders in perturbation theory.

The 𝒩 = 4 coset model and the higher spin algebra

By computing the operator product expansions between the first two [Formula: see text] higher spin multiplets in the unitary coset model, the (anti-)commutators of higher spin currents are obtained under the large [Formula: see text] ’t Hooft-like limit. The free field realization with complex bosons and fermions is presented. The (anti-)commutators for generic spins [Formula: see text] and [Formula: see text] with manifest [Formula: see text] symmetry at vanishing ’t Hooft-like coupling constant are completely determined. The structure constants can be written in terms of the ones in the [Formula: see text] [Formula: see text] algebra found by Bergshoeff, Pope, Romans, Sezgin and Shen previously, in addition to the spin-dependent fractional coefficients and two [Formula: see text] invariant tensors. We also describe the [Formula: see text] higher spin generators, by using the above coset construction results, for general superspin [Formula: see text] in terms of oscillators in the matrix generalization of [Formula: see text] Vasiliev higher spin theory at nonzero ’t Hooft-like coupling constant. We obtain the [Formula: see text] higher spin algebra for low spins and present how to determine the structure constants, which depend on the higher spin algebra parameter, in general, for fixed spins [Formula: see text] and [Formula: see text].

Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero

This paper is a continuation of [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342]. We present certain new applications and generalizations of the free field realization of the twisted Heisenberg–Virasoro algebra [Formula: see text] at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for [Formula: see text] is presented by combining a bosonic construction of Whittaker modules from [D. Adamović, R. Lu and K. Zhao, Whittaker modules for the affine Lie algebra [Formula: see text], Adv. Math. 289 (2016) 438–479; arXiv:1409.5354] with a construction of logarithmic modules for vertex algebras. As an application, we prove that there exists a non-split self-extension of irreducible self-dual module which is a logarithmic module of rank two. We construct a large family of logarithmic modules containing different types of highest weight modules as subquotients. We believe that these logarithmic modules are related with projective covers of irreducible modules in a suitable category of [Formula: see text]-modules.

Casimir 𝒲𝒜N algebras as the truncated 𝒲∞ algebra

The complete structure of the Casimir [Formula: see text] algebras is shown to exist in such a way that the Casimir [Formula: see text] algebra is a kind of truncated type of [Formula: see text] algebra both in the primary and in the quadratic basis, first using the associativity conditions in the basis of primary fields and second using the Miura basis coming from the free field realization as a different basis. We can conclude that the Casimir [Formula: see text] algebra is a kind of truncated type of [Formula: see text] algebra, so it is clear from any construction of [Formula: see text] algebra that by putting infinite number of fields [Formula: see text] with [Formula: see text] to zero, we arrive at the Casimir [Formula: see text] algebra. We concentrated in this work only for the particular case of [Formula: see text] algebra since this example gives us explicitly a method on how to deal with the general case [Formula: see text].