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A<sc>bstract</sc> In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions, extending our previous work. Specifically, in this work we discuss more general gauged noninvertible symmetries in which the noninvertible symmetry is not multiplicity free, and discuss the case of Rep(A₄) in detail. We realize Rep(A₄) gaugings for thec= 1 CFT at the exceptional point in the moduli space and find new self-duality under gauging a certain non-group algebra object, leading to a larger noninvertible symmetry Rep(SL(2, ℤ₃)). We also discuss more general examples of decomposition in two-dimensional gauge theories with trivially-acting gauged noninvertible symmetries. 

A<sc>bstract</sc> In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form$$ \textrm{Rep}\left(\mathcal{H}\right) $$ $\mathrm{Rep}\left(H\right)$for$$ \mathcal{H} $$ $H$a suitable Hopf algebra (which includes the special case Rep(G) forGa finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on$$ {\mathcal{H}}^{\ast } $$ $H^{\ast }$. We discuss how ordinaryGorbifolds for finite groupsGare a special case of the construction, corresponding to the fusion category Vec(G) = Rep(ℂ[G]^*). For the cases Rep(S₃), Rep(D₄), and Rep(Q₈), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep(S₃), Rep(D₄), Rep(Q₈), and$$ \textrm{Rep}\left({\mathcal{H}}_8\right) $$ $\mathrm{Rep}\left(H_8\right)$, and discuss applications inc= 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially. 

Monitoring water quality by detecting chemical and biological contaminants is critical to ensuring the provision and discharge of clean water, hence protecting human health and the ecosystem. Among the available analytical techniques, infrared (IR) spectroscopy provides sensitive and selective detection of multiple water contaminants. In this work, we present an application of IR spectroscopy for qualitative and quantitative assessment of chemical and biological water contaminants. We focus on in-line detection of nitrogen pollutants in the form of nitrate and ammonium for wastewater treatment process control and automation. We discuss the effects of water quality parameters such as salinity, pH, and temperature on the IR spectra of nitrogen pollutants. We then focus on application of the sensor for detection of contaminants of emerging concern, such as arsenic and Per- and polyfluoroalkyl substances (PFAS) in drinking water. We demonstrate the use of multivariate statistical analysis for automated data processing in complex fluids. Finally, we discuss application of IR spectroscopy for detecting biological water contaminants. We use the metabolomic signature of E. coli bacteria to determine its presence in water as well as distinguish between different strains of bacteria. Overall, this work shows that IR spectroscopy is a promising technique for monitoring both chemical and biological contaminants in water and has the potential for real-time, inline water quality monitoring.