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An admissible level \(\widehat{\mathfrak {osp}} \left( 1 \big \vert 2 \right) \)-model: modular transformations and the Verlinde formula

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Abstract

The modular properties of the simple vertex operator superalgebra associated with the affine Kac–Moody superalgebra \(\widehat{{\mathfrak {osp}}} (1|2)\) at level \(-\frac{5}{4}\) are investigated. After classifying the relaxed highest-weight modules over this vertex operator superalgebra, the characters and supercharacters of the simple weight modules are computed and their modular transforms are determined. This leads to a complete list of the Grothendieck fusion rules by way of a continuous superalgebraic analog of the Verlinde formula. All Grothendieck fusion coefficients are observed to be non-negative integers. These results indicate that the extension to general admissible levels will follow using the same methodology once the classification of relaxed highest-weight modules is completed.

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Notes

  1. We mention that the corresponding analysis for the \(\widehat{{\mathfrak {gl}}} (1|1)\) logarithmic conformal field theory, carried out in [29], was restricted to the Neveu–Schwarz sector as the simple characters of this sector closed on themselves under modular transformations. The same is not true for \(\widehat{{\mathfrak {osp}}} (1|2)\) conformal field theories.

  2. We assume throughout that when \(\lambda \) solves the reducibility condition, then it is the unique element of \(\Lambda \) that does so. This need not be the case, as for certain q there are two solutions for \(\lambda \) in \(\Lambda \). However, the corresponding reducible modules will, again, not be needed here.

  3. The analogous analysis in which \(v_{\mu }\) has odd parity leads to equivalent constraints on s.

  4. This is actually non-trivial to prove rigorously and we shall not do so here, instead referring to [73] for the details.

  5. We remark that our working definition of module of a vertex superalgebra follows that given by Frenkel and Ben-Zvi [78, Ch. 5.1], adding the requirement to be \({\mathbb {Z}}_2\)-graded by parity (as in Sect. 2.2). In particular, a module over the level-k universal vertex superalgebra associated with \({\mathfrak {osp}} (1|2)\) is just a smooth \({\mathbb {Z}}_2\)-graded level-k \(\widehat{{\mathfrak {osp}}} (1|2)\)-module.

  6. For v of definite conformal weight \(\Delta _v\), we assume a mode expansion for the corresponding field of the form \(v(z) = \sum _n v_n z^{-n-\Delta _v}\).

  7. Recall that in rings of formal Laurent series, the powers of the indeterminate are assumed to be bounded below.

  8. Strictly speaking, the material in [94] assumes the setting of a rational vertex operator algebra. However, it is possible to generalise many of their results to the standard module formalism, see [95] for details.

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Acknowledgements

DR thanks Kenji Iohara for illuminating discussions on the structure of Verma modules over \(\widehat{{\mathfrak {osp}}} (1|2)\). We would like to thank the anonymous referee whose careful reading of the original manuscript and many suggestions significantly improved the article. JS’s research is supported by a University Research Scholarship from the Australian National University. DR’s research is supported by the Australian Research Council Discovery Projects DP1093910 and DP160101520 as well as the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers CE140100049. SW’s research is supported by Australian Research Council Discovery Early Career Researcher Award DE140101825 and the Australian Research Council Discovery Project DP160101520.

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Snadden, J., Ridout, D. & Wood, S. An admissible level \(\widehat{\mathfrak {osp}} \left( 1 \big \vert 2 \right) \)-model: modular transformations and the Verlinde formula. Lett Math Phys 108, 2363–2423 (2018). https://doi.org/10.1007/s11005-018-1097-5

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