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Let V V be a vertex operator superalgebra with the natural order 2 automorphism σ \sigma . Under suitable conditions on V V , the σ \sigma -fixed subspace V 0 ¯ V_{\bar 0} is a vertex operator algebra and the V 0 ¯ V_{\bar 0} -module category C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a modular tensor category. In this paper, we prove that C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a fermionic modular tensor category and the Müger centralizer C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 of the fermion in C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is generated by the irreducible V 0 ¯ V_{\bar 0} -submodules of the V V -modules. In particular, C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 is a super-modular tensor category and C V 0 ¯ \mathcal {C}_{V_{\bar 0}} is a minimal modular extension of C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 . We provide a construction of a vertex operator superalgebra V l V^l for each positive integer l l such that C ( V l ) 0 ¯ \mathcal {C}_{{(V^l)_{\bar 0}}} is a minimal modular extension of C V 0 ¯ 0 \mathcal {C}_{V_{\bar 0}}^0 . We prove that these modular tensor categories C ( V l ) 0 ¯ \mathcal {C}_{{(V^l)_{\bar 0}}} are uniquely determined, up to equivalence, by the congruence class of l l modulo 16. 

Abstract To complement velocity distributions, seismic attenuation provides additional important information on fluid properties of hydrocarbon reservoirs in exploration seismology, as well as temperature distributions, partial melting, and water content within the crust and mantle in earthquake seismology. Full waveform inversion (FWI), as one of the state‐of‐the‐art seismic imaging techniques, can produce high‐resolution constraints for subsurface (an)elastic parameters by minimizing the difference between observed and predicted seismograms. Traditional waveform inversion for attenuation is commonly based on the standard‐linear‐solid (SLS) wave equation, in which case the quality factor (Q) has to be converted to stress and strain relaxation times. When using multiple attenuation mechanisms in the SLS method, it is difficult to directly estimate these relaxation time parameters. Based on a time domain complex‐valued viscoacoustic wave equation, we present an FWI framework for simultaneously estimating subsurfacePwave velocity and attenuation distributions. BecauseQis explicitly incorporated into the viscoacoustic wave equation, we directly derivePwave velocity andQsensitivity kernels using the adjoint‐state method and simultaneously estimate their subsurface distributions. By analyzing the Gauss‐Newton Hessian, we observe strong interparameter crosstalk, especially the leakage from velocity toQ. We approximate the Hessian inverse using a preconditioned L‐BFGS method in viscoacoustic FWI, which enables us to successfully reduce interparameter crosstalk and produce accurate velocity and attenuation models. Numerical examples demonstrate the feasibility and robustness of the proposed method for simultaneously mapping complex velocity andQdistributions in the subsurface.