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Abstract We introduce a distributional Jacobian determinant \det DV_{\beta}(Dv)in dimension two for the nonlinear complex gradient V_{\beta}(Dv)=\lvert Dv\rvert^{\beta}(v_{x_{1}},-v_{x_{2}})for any \beta>-1, whenever v\in W^{1\smash{,}2}_{\mathrm{loc}}and \beta\lvert Dv\rvert^{1+\beta}\in W^{1\smash{,}2}_{\mathrm{loc}}.This is new when \beta\neq 0.Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant \det DV_{\beta}(Du)is a nonnegative Radon measure with some quantitative local lower and upper bounds.We also give the following two applications. Applying this result with \beta=0, we develop an approach to build up a Liouville theorem, which improves that of Savin.Precisely, if 𝑢 is an ∞-harmonic function in the whole \mathbb{R}^{2}with \liminf_{R\to\infty}\inf_{c\in\mathbb{R}}\frac{1}{R}\barint_{B(0,R)}\lvert u(x)-c\rvert\,dx<\infty,then u=b+a\cdot xfor some b\in\mathbb{R}and a\in\mathbb{R}^{2}.Denoting by u_{p}the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that \det DV_{\beta}(Du_{p})\to\det DV_{\beta}(Du)as p\to\inftyin the weak-⋆ sense in the space of Radon measure.Recall that V_{\beta}(Du_{p})is always quasiregular mappings, but V_{\beta}(Du)is not in general.
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This content will become publicly available on August 17, 2025
Ordinary modules for vertex algebras of 𝔬𝔰𝔭 1|2𝑛
Abstract We show that the affine vertex superalgebra V^{k}(\mathfrak{osp}_{1|2n})at generic level 𝑘 embeds in the equivariant 𝒲-algebra of \mathfrak{sp}_{2n}times 4nfree fermions.This has two corollaries:(1) it provides a new proof that, for generic 𝑘, the coset \operatorname{Com}(V^{k}(\mathfrak{sp}_{2n}),V^{k}(\mathfrak{osp}_{1|2n}))is isomorphic to \mathcal{W}^{\ell}(\mathfrak{sp}_{2n})for \ell=-(n+1)+(k+n+1)/(2k+2n+1), and(2) we obtain the decomposition of ordinary V^{k}(\mathfrak{osp}_{1|2n})-modules into V^{k}(\mathfrak{sp}_{2n})\otimes\mathcal{W}^{\ell}(\mathfrak{sp}_{2n})-modules.Next, if 𝑘 is an admissible level and ℓ is a non-degenerate admissible level for \mathfrak{sp}_{2n}, we show that the simple algebra L_{k}(\mathfrak{osp}_{1|2n})is an extension of the simple subalgebra L_{k}(\mathfrak{sp}_{2n})\otimes{\mathcal{W}}_{\ell}(\mathfrak{sp}_{2n}).Using the theory of vertex superalgebra extensions, we prove that the category of ordinary L_{k}(\mathfrak{osp}_{1|2n})-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects.It is equivalent to a certain subcategory of \mathcal{W}_{\ell}(\mathfrak{sp}_{2n})-modules.A similar result also holds for the category of Ramond twisted modules.Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary L_{k}(\mathfrak{sp}_{2n})-modules are rigid.
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- Award ID(s):
- 2001484
- PAR ID:
- 10558391
- Publisher / Repository:
- De Gruyter
- Date Published:
- Journal Name:
- Journal für die reine und angewandte Mathematik (Crelles Journal)
- Volume:
- 817
- ISSN:
- 0075-4102
- Page Range / eLocation ID:
- 1-31
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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