Abstract
A family of quantum fields is said to be strongly local if it generates a local net of von Neumann algebras. There are few methods of showing directly strong locality of a quantum field. Among them, linear energy bounds are the most widely used, yet a chiral conformal field of conformal weight \(d>2\) cannot admit linear energy bounds. In this paper we give a new direct method to prove strong locality in two-dimensional conformal field theory. We prove that if a chiral conformal field satisfies an energy bound of degree \(d-1\), then it also satisfies a certain local version of the energy bound, and this in turn implies strong locality. A central role in our proof is played by diffeomorphism symmetry. As a concrete application, we show that the vertex operator algebra given by a unitary vacuum representation of the \({\mathcal {W}}_3\)-algebra is strongly local. For central charge \(c > 2\), this yields a new conformal net. We further prove that these nets do not satisfy strong additivity, and hence are not completely rational.
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22 July 2023
A Correction to this paper has been published: https://doi.org/10.1007/s00220-023-04791-w
Notes
This assumption is mainly for simplicity. In general, for applications of e.g. Proposition 3.2—since the field operators always change the eigenvalues by an integer—one could decompose the space V into invariant components where this assumption does hold and then use the fact that the estimate of the cited proposition is independent of the value of h.
The constant r is given in an explicit manner in [FH05, (3.27)]: \(r = -\frac{c}{24}\int _{S^1}\{\gamma , z\}f(z)dz\), where \(\{\gamma , z\}\) is the Schwarz derivative.
This can be proved from the linear enegy bound and induction: The \(n=1\) case is exactly the linear energy bound. Then, \(\Vert T(f_1)\cdots T(f_n) \Psi \Vert \le t\Vert (L_0+{\mathbb {1}})T(f_2)\cdots T(f_n) \Psi \Vert \) and \((L_0+{\mathbb {1}})T(f_2)\cdots T(f_n) \Psi = \sum _{j=2}^n T(f_2)\cdots T(f_j')\cdots T(f_n)\Psi + T(f_2)\cdots T(f_n) (L_0+{\mathbb {1}})\Psi \), and we can apply the assumption of induction to the last expression, as \(\Psi \in C^\infty (L_0)\). For the second claim, we take \(n > s\), then we have \((T(f)+q{\mathbb {1}})^{2n} \le t(L_0+{\mathbb {1}})^{2n}\) for some \(t>0\), and as the function \(f(x) = x^{\frac{s}{n}}\) is an operator monotone [Sim19, Theorem 4.1], we have \((T(f)+q{\mathbb {1}})^{2s} \le (t(L_0+{\mathbb {1}}))^{2s}\) by [Sim19, Theorem 2.9].
Not all nonnegative smooth functions have smooth \((d-1)\)-th root.
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Acknowledgements
We would like to thank the referees for helpful comments. SC and MW are supported in part by the ERC advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”. SC is also supported by GNAMPA-INDAM. YT was supported until February 2020 by Programma per giovani ricercatori, anno 2014 “Rita Levi Montalcini” of the Italian Ministry of Education, University and Research. MW is supported also by the National Research, Development and Innovation Office of Hungary (NRDI) via the research Grant K124152, KH129601 and K132097 and the Bolyai János Fellowship of the Hungarian Academy of Sciences, the ÚNKP-21-5 New National Excellence Program of the Ministry for Innovation and Technology. SC and YT acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome “Tor Vergata” CUP E83C18000100006 and the University of Rome “Tor Vergata” funding scheme “Beyond Borders” CUP E84I19002200005. We are also grateful to Mathematisches Forschungsinstitut Oberwolfach and Simons Center for Geometry and Physics, where parts of this work have been done.
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Carpi, S., Tanimoto, Y. & Weiner, M. Local Energy Bounds and Strong Locality in Chiral CFT. Commun. Math. Phys. 390, 169–192 (2022). https://doi.org/10.1007/s00220-021-04291-9
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DOI: https://doi.org/10.1007/s00220-021-04291-9