Modular invariance of (logarithmic) intertwining operators
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915511816683520 |
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| author | Huang, Yi-Zhi |
| author_facet | Huang, Yi-Zhi |
| contents | Let $V$ be a $C_2$-cofinite vertex operator algebra without nonzero elements of negative weights. We prove the conjecture that the spaces spanned by analytic extensions of pseudo-$q$-traces ($q=e^{2πiτ}$) shifted by $-\frac{c}{24}$ of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized $V$-modules are invariant under modular transformations. The convergence and analytic extension result needed to formulate this conjecture and some consequences on such shifted pseudo-$q$-traces were proved by Fiordalisi in [F1] and [F2] using the method developed in [H2]. The method that we use to prove this conjecture is based on the theory of the associative algebras $A^{N}(V)$ for $N\in \mathbb{N}$, their graded modules and their bimodules introduced and studied by the author in [H8] and [H9]. This modular invariance result gives a construction of $C_2$-cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_15152 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Modular invariance of (logarithmic) intertwining operators Huang, Yi-Zhi Quantum Algebra High Energy Physics - Theory Rings and Algebras Representation Theory 17B69, 81T40, 32N10 Let $V$ be a $C_2$-cofinite vertex operator algebra without nonzero elements of negative weights. We prove the conjecture that the spaces spanned by analytic extensions of pseudo-$q$-traces ($q=e^{2πiτ}$) shifted by $-\frac{c}{24}$ of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized $V$-modules are invariant under modular transformations. The convergence and analytic extension result needed to formulate this conjecture and some consequences on such shifted pseudo-$q$-traces were proved by Fiordalisi in [F1] and [F2] using the method developed in [H2]. The method that we use to prove this conjecture is based on the theory of the associative algebras $A^{N}(V)$ for $N\in \mathbb{N}$, their graded modules and their bimodules introduced and studied by the author in [H8] and [H9]. This modular invariance result gives a construction of $C_2$-cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories. |
| title | Modular invariance of (logarithmic) intertwining operators |
| topic | Quantum Algebra High Energy Physics - Theory Rings and Algebras Representation Theory 17B69, 81T40, 32N10 |
| url | https://arxiv.org/abs/2305.15152 |