Narrow Results

Induced quantum gravity dynamics built over a Riemann surface is studied in arbitrary dimension. Local coordinates on the target space are given by means of the Laguerre–Forsyth construction. A simple model is proposed and perturbatively quantized. In doing so, the classical -symmetry turns out to be preserved on-shell at any order of the ℏ perturbative expansion. As a main result, due to quantum corrections, the target coordinates acquire a nontrivial character.

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra $W^0{(p)}_{A_n}$, $1\le n\le 3$, and from $Ẑ$-invariants of three-manifolds associated with gauge group SU(3).

Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the ${𝔰𝔩}_{n+1}$ subregular W-algebra can be realized in terms of the ${𝔰𝔩}_{n+1}$ regular W-algebra and the half lattice vertex algebra $\mathrm{\Pi }$. This generalizes the realizations found for $n=1$ and $2$ in [D. Adamović, Realizations of simple affine vertex algebras and their modules: The cases $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$, Comm. Math. Phys. 366 (2019) 1025–1067, arXiv:1711.11342 [math.QA]; D. Adamović, K. Kawasetsu and D. Ridout, A realization of the Bershadsky–Polyakov algebras and their relaxed modules, Lett. Math. Phys., 111 (2021) 1–30, arXiv:2007.00396 [math.QA]] and can be interpreted as an inverse quantum hamiltonian reduction in the sense of Adamović. We use this realization to explore the representation theory of ${𝔰𝔩}_{n+1}$ subregular W-algebras. Much of the structure encountered for ${𝔰𝔩}_2$ and ${𝔰𝔩}_3$ is also present for ${𝔰𝔩}_{n+1}$. Particularly, the simple ${𝔰𝔩}_{n+1}$ subregular W-algebra at nondegenerate admissible levels can be realized purely in terms of the $W_{n+1}$ minimal model vertex algebra and $\mathrm{\Pi }$.

We prove that when Kontsevich’s deformation quantization is applied on weight homogeneous Poisson structures, the operators in the ∗-product formula are weight homogeneous. In the linear Poisson case for a semisimple Lie algebra $𝔤$ the Poisson manifold $X$ is ${𝔤}^{\ast }$. As an application we provide an isomorphism between the Cattaneo–Felder–Torossian reduction algebra $H^0({𝔪}^{\perp },\chi ,𝔮)$ and the $W$-algebra ${(U(𝔤)/U(𝔤){𝔪}_{\chi })}^{𝔪}$. We also show that in the $W$-algebra setting, ${(S(𝔤)/S(𝔤){𝔪}_{\chi })}^{𝔪}$ is polynomial. Finally, we compute generators of $H^0({𝔪}^{\perp },\chi ,𝔮)$ as a deformation of ${(S(𝔤)/S(𝔤){𝔪}_{\chi })}^{𝔪}$.

We study the representation theory of the Bershadsky–Polyakov algebra ${{𝒲}}_k={{𝒲}}_k(s{l}_3,f_{𝜃})$. In particular, Zhu algebra of ${{𝒲}}_k$ is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector. We classify all modules in the category ${𝒪}$ for the Bershadsky–Polyakov algebra ${{𝒲}}_k$ for $k=-5/3,-9/4,-1,0$. In the case $k=0$, we show that the Zhu algebra $A({{𝒲}}_k)$ has two-dimensional indecomposable modules.

In this study, we study principal admissible representations for the affine Lie superalgebra $𝔬𝔰𝔭{(1|2n)}^{ˆ}$. Using the character formula of irreducible admissible representations of $𝔬𝔰𝔭{(1|2n)}^{ˆ}$, we calculate a character formula of $W$-modules which are obtained from the quantized Drinfeld–Sokolov reduction and principal admissible representations. As a by-product, we obtain the minimal series modules of the Neveu–Schwarz algebra through the $W$-modules arising from the principal admissible modules over $𝔬𝔰𝔭{(1|2)}^{ˆ}$.