Rational and non-rational two-dimensional conformal field theories arising from lattices
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| Format: | Preprint |
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2025
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| _version_ | 1866913870001471488 |
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| author | Adamo, Maria Stella Giorgetti, Luca Tanimoto, Yoh |
| author_facet | Adamo, Maria Stella Giorgetti, Luca Tanimoto, Yoh |
| contents | For a (finite-dimensional) real Hilbert space $\mathfrak h$ and an orthogonal projection $p$, we consider the associated Heisenberg Lie algebra and the two-dimensional Heisenberg conformal net. Given an even lattice $Q$ in $\mathfrak h$ with respect to the indefinite bilinear form on $\mathfrak h$ defined by $p$, we construct a two-dimensional conformal net ${\mathcal A}_Q$ extending the Heisenberg conformal net. Moreover, with a certain discreteness assumption on the spectrum of the extension, we show that any two-dimensional extension of the Heisenberg conformal net is of the form ${\mathcal A}_Q$ up to unitary equivalence.
We consider explicit examples of even lattices where $\mathfrak h$ is two-dimensional and $p$ is one-dimensional, and we show that the extended net may have completely rational or non-completely rational chiral (i.e. one-dimensional lightray) components, depending on the choice of lattice. In the non-rational case, we exhibit the braided equivalence of a certain subcategory of the representation category of the chiral Heisenberg net corresponding to the two-dimensional lattice extension.
Inspired by the charge and braiding structures of these nets, we construct two-dimensional conformal Wightman fields on the same Hilbert spaces. We show that, in some cases, these Wightman fields generate the corresponding extended nets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_01008 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational and non-rational two-dimensional conformal field theories arising from lattices Adamo, Maria Stella Giorgetti, Luca Tanimoto, Yoh Mathematical Physics Operator Algebras Representation Theory 81T05, 81T40, 46L60 For a (finite-dimensional) real Hilbert space $\mathfrak h$ and an orthogonal projection $p$, we consider the associated Heisenberg Lie algebra and the two-dimensional Heisenberg conformal net. Given an even lattice $Q$ in $\mathfrak h$ with respect to the indefinite bilinear form on $\mathfrak h$ defined by $p$, we construct a two-dimensional conformal net ${\mathcal A}_Q$ extending the Heisenberg conformal net. Moreover, with a certain discreteness assumption on the spectrum of the extension, we show that any two-dimensional extension of the Heisenberg conformal net is of the form ${\mathcal A}_Q$ up to unitary equivalence. We consider explicit examples of even lattices where $\mathfrak h$ is two-dimensional and $p$ is one-dimensional, and we show that the extended net may have completely rational or non-completely rational chiral (i.e. one-dimensional lightray) components, depending on the choice of lattice. In the non-rational case, we exhibit the braided equivalence of a certain subcategory of the representation category of the chiral Heisenberg net corresponding to the two-dimensional lattice extension. Inspired by the charge and braiding structures of these nets, we construct two-dimensional conformal Wightman fields on the same Hilbert spaces. We show that, in some cases, these Wightman fields generate the corresponding extended nets. |
| title | Rational and non-rational two-dimensional conformal field theories arising from lattices |
| topic | Mathematical Physics Operator Algebras Representation Theory 81T05, 81T40, 46L60 |
| url | https://arxiv.org/abs/2506.01008 |