A braided monoidal category for free super-bosons. (English) Zbl 1405.81138
Summary: The chiral conformal field theory of free super-bosons is generated by weight one currents whose mode algebra is the affinisation of an abelian Lie super-algebra \(\mathfrak{h}\) with non-degenerate super-symmetric pairing. The mode algebras of a single free boson and of a single pair of symplectic fermions arise for even|odd dimension \(1|0\) and \(0|2\) of \(\mathfrak{h}\), respectively. In this paper, the representations of the untwisted mode algebra of free super-bosons are equipped with a tensor product, a braiding, and an associator. In the symplectic fermion case, i.e., if \(\mathfrak {h}\) is purely odd, the braided monoidal structure is extended to representations of the \(\mathbb {Z}/2\mathbb {Z}\)-twisted mode algebra. The tensor product is obtained by computing spaces of vertex operators. The braiding and associator are determined by explicit calculations from three- and four-point conformal blocks.{
©2014 American Institute of Physics}
©2014 American Institute of Physics}
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 17A70 | Superalgebras |
| 81R25 | Spinor and twistor methods applied to problems in quantum theory |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
References:
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| [2] | There is a potential pitfall with the above definition of \documentclass[12pt]{minimal}\( \begin{document}\overline{\operatorname{Ind}}(R)\end{document}\) which originates from using the H-grading instead of the (in general non-existent) \(L_0\)-grading. Namely, consider the case \documentclass[12pt]{minimal}\( \begin{document}\mathfrak{h}= \mathbb{C}^{1|0}\end{document}\) and \documentclass[12pt]{minimal}\( \begin{document}R = \bigoplus_{m \in \mathbb{Z}} \mathbb{C}_m\end{document} \), where \documentclass[12pt]{minimal}\( \begin{document}\mathbb{C}_m\end{document}\) is the one-dimensional \documentclass[12pt]{minimal}\( \begin{document}\mathfrak{h}\end{document} \)-module on which the generator of \documentclass[12pt]{minimal}\( \begin{document}\mathfrak{h}\end{document}\) acts by multiplication by m. Then the vector \documentclass[12pt]{minimal}\( \begin{document}1_m \in \mathbb{C}_m\end{document}\) has \(L_0\)-eigenvalue \documentclass[12pt]{minimal}\( \begin{document}\frac{1}{2} m^2\end{document}\) and H-eigenvalue 0. The infinite sum \documentclass[12pt]{minimal}\( \begin{document}\sum_{m \in \mathbb{Z}} 1_m\end{document}\) is therefore an element of the algebraic completion of \documentclass[12pt]{minimal}\( \begin{document}\operatorname{Ind}(R)\end{document}\) with respect to the \(L_0\)-gradation, but it is not in the completion with respect to the H-gradation, i.e., \documentclass[12pt]{minimal}\( \begin{document}\sum_{m \in \mathbb{Z}} 1_m \notin \overline{\operatorname{Ind}}(R)\end{document} \). Since the \(L_0\)-gradation is often used in conformal field theory, this point should be kept in mind. |
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